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FOUNDED  BY  JOHN  D.  ROCKEFELLER 


ON  THE  RESOLUTION  OF  HIGHER 

SINGULARITIES  OF  ALGEBRAIC 

CURVES  INTO  ORDINARY  NODES 


BY 


B.  M.  WALKER 


A   DISSERTATION 

Submitted  to  the  Faculties  of  the  Graduate  Schools  op  Arts, 

Literature,  and  Science  in  Candidacy  for  the 

Degree  of  Doctor  of  Philosophy 


DEPARTMENT   OF   MATHEMATICS 


CHICAGO 

1906 


V 


Zbc  TElntvexsity  of  Cbtcaao 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 


ON  THE  RESOLUTION  OF  HIGHER 

SINGULARITIES  OF  ALGEBRAIC 

CURVES  INTO  ORDINARY  NODES 


BY 


B.  M.  WALKER 


A   DISSERTATION 

Submitted  to  the  Faculties  of  the  Graduate  Schools  of  Arts, 

Literature,  and  Science  in  Candidacy  for  the 

Degree  of  Doctor  of  Philosophy 


DEPARTMENT   OF  MATHEMATICS 


CHICAGO 
1906 


PREftS  OF 

The  New  Era  printing  Company 

lancaster,  pa. 


ON  THE   RESOLUTION    OF    HIGHER    SINGULARITIES   OF 
ALGEBRAIC   CURVES   INTO   ORDINARY  NODES. 

Introduction. 

Every  algebraic  curve  C  can  be  transformed  by  a  birational 
transformation  of  the  plane  (a  so-called  Cremona  transformation)  into 
a  curve  C  which  has  no  other  singular  points  than  ordinary  *  mul- 
tiple points.  This  fundamental  theorem  was  established  by  M. 
NoETHERf  in  1875. 

By  a  birational  transformation  of  the  curve  (a  so-called  Riemann 
transformation),  the  curve  Cf  can  further  be  transformed  into  a 
curve  C"  which  has  no  other  singular  points  than  ordinary  *  double 
points.  It  seems  that  this  theorem  was  first  enunciated  by 
Halphen.  J 

Three  essentially  different  methods  have  been  employed  for  its 
proof: 

1)  Picard  §  uses  a  quadratic  one-to-three  transformation  of  the 
plane. 

2)  Bertini  II  uses,  for  the  same  purpose,  a  cubic  one-to-two  trans- 
formation of  the  plane. 

*  By  an  ordinary  multiple  point,  we  mean,  in  the  seqnel,  a  mnltiple  point  all  of 
whose  cyoles  are  linear  and  have  distinct  tangents.  For  the  definition  of  cycles, 
see  Halphen,  "Traite  de  G4om4trie  de  Salmon,"  p.  541,  and  Jordan,  "Cours 
D'Analyse,"  t.  I,  p.  563. 

t  Noktheb,  u  Ueber  die  singnlaren  Werthsysteme  einer  algebraischen  Function 
nnd  die  singnlaren  Pnnkte  einer  algebraischen  ourve,"  Mathematische  Annalen,  Bd. 
9,  p.  166.  Compare  also,  Picard,  "Traite  D' Analyse,"  t.  II,  p.  364,  Jordan, 
"  Conrs  D' Analyse,"  1. 1,  p.  588,  and  Halphen,  "  Traite*  de  G6ome*trie  de  Salmon," 
p.  630. 

X  Halphen,  ' '  Traite  de  Geom^trie  de  Salmon, » »  pp.  630,  631,  632.  His  proof  is, 
however,  incomplete,  as  has  been  pointed  ont  by  Picard,  "Traite  D'Analyse,"  t. 
II,  p.  366. 

§  Picard,  "Traite*  D'Analyse,"  t.  II,  p.  366,  1893,  and  Simart,  "Snr  un 
the\)reme  relatif  a  la  transformation  des  courbes  algSbriques, "  Comptes  rendus  de 
VAcademie,  t.  116,  p.  1047,  1893. 

||  Bertini,  "  Trasformazione  di  una  cnrva  algebrica  in  nn'altra  con  soli  pnnti 
doppi,"  Mathematische  Annalen,  Bd.  44,  p.  158,  1894. 

3 


166202 


4  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

3)  Polncar£  *  transforms  the  plane  curve  into  a  curve  in  space, 
which  has  no  singular  points  ;  and  then  shows  that  this  curve  may 
be  projected  into  a  plane  curve  with  no  other  singular  points  than 
ordinary  double  points.  All  these  proofs,  however,  are  written  in 
an  exceedingly  concise  style  and  leave  a  great  many  minor  points  to 
the  reader. 

The  object  of  the  present  paper  is  to  give  a  new  proof  of  the 
theorem  in  question  by  carrying  out,  in  detail,  an  idea  contained  in 
the  following  foot-note  added  by  Professor  Felix  Klein  to  Bertini's 
paper  : 

"  Die  Methode  von  Bertini  kommt  geometrisch  zu  reden  darauf 
zuruck,  die  Ebene,  in  welcher  uns  die  Curve  mit  siDgularen  Punkte 
gegeben  ist,  als  eindeutige  Abbildung  einer  Flache  3.  Ordnung  zu 
betrachten,  dadurch  die  Curve  in  eine  Raumcurve  zu  verwandeln  und 
letztere  hinterher  wieder  von  einen  hinreichend  allgemeinen  Punkte 
aus  auf  eine  andere  Ebene  zu  projiciren.  In  dieser  Form  ist  mir  der 
Ansatz  noch  von  Clebsch  her  bekannt,  der  mir  denselben  in  Herbst 
1869  mundlich  mittheilte." 

The  paper  is  divided  into  four  chapters.  In  Chapter  I,  those  prop- 
erties of  triply  infinite  systems  of  cubics  are  established,  which  are 
needed  for  the  transformation  from  the  plane  to  the  cubic  surface. 
Chapters  II  and  III  deal  with  the  transformation  from  the  plane  to 
the  cubic  surface,  f  and  its  effect  upon  the  given  algebraic  curve. 
Let  A19  A2 ,  •  •  • ,  Ar  be  the  singular  points  of  the  given  algebraic 
curve  -ST in  the  original  plane  II ;  they  are  all  supposed  to  be  ordinary 
multiple  points.  It  is  proved  that  the  six  fundamental  points  of  the 
transformation  can  always  be  so  chosen  that  one  of  the  multiple 
points  of  K,  say  Al}  is  resolved  into  simple  points  of  the  image 
K'  of  K  on  the  cubic  surface  and  all  the  remaining  multiple  points 
A2,  •  •  •,  Ar  of  iTare  transformed  into  new  ordinary  J  multiple  points 


*Poincabk,  "Sur  les  transformations  birationelles  des  courbes  alg^briques," 
Comptes  rendus  de  VAcademie,  t.  117,  p.  18,  1893.  For  additional  literature  see 
"  Encyklopadie  der  Mathematisehen  Wissenschaften,"  Band  III, ,  Heft  3,  1906. 

fClebscb's  well-known  one-to-one  correspondence  between  plane  and  cubic  sur- 
face ("Geometrie  auf  den  Flachen  dritter  Ordnung,"  Crelle,  65). 

t  Each  oneof  these  points  isthecenterof  a  finite  number  of  linear  cycles  (Jordan, 
"Cours  D' Analyse,"  t.  I,  p.  563)  with  distinct  tangents. 


INTRODUCTION.  5 

A'2j  •  •  •,  A'r  of  K'  unchanged  in  orders  of  multiplicity  and  with  dis- 
tinct tangents. 

In  Chapter  IV,  the  curve  K'  is  projected  into  a  new  curve  K" ',  in 
a  plane  II',  from  a  center  of  projection  0  chosen  on  the  cubic  surface. 
It  is  proved  that  the  point  0  can  always  be  so  chosen  that  the  curve 
K"  possesses  one  multiple  point  less  than  the  curve  K  and  has  all 
the  remaining  multiple  points  A2,  •  •  •,  Ar  of  K  transformed  into  new 
ordinary  multiple  points  A't ,  •  •  • ,  A'r  unchanged  in  orders  of  mul- 
tiplicity and  with  distinct  tangents,  and  a  finite  number  of  additional 
ordinary  double  points,  but  no  other  singular  points. 

A  finite  number  of  repetitions  of  this  process  leads  to  a  curve 
which  has  no  other  singular  points  than  ordinary  nodes. 


CHAPTER   I. 

Properties  of  Triply   Infinite  Linear  Systems  of   Plane 

Cubics. 

§1.   Triply  infinite  linear  systems  of  cubics. 

We  shall  develop,  in  this  chapter,  a  number  of  auxiliary  theorems 
on  triply  infinite  linear  systems  of  cubics,  which  will  be  needed  in 
our  discussion  of  the  one-to-one  correspondence  between  the  plane 
and  the  cubic  surface. 

In  a  plane  II,  whose  points  are  referred  to  a  system  of  trilinear 
coordinates  x:y  :z ,  there  are  given  six  distinct  points  P^x^  yif  z.), 
(t  a*  1,  2,  •••,  6),  chosen  so  that  no  three  points  lie  on  a  straight  line. 
We  consider  the  problem  : 

To  determine  all  cubics  passing  through  the  six  points. 
Let 

/  =  Ax3  +  By3  +  Cz*  +  Dx2y  +  Erfz  +  Fxtf 

\  +  Gxz2  +  Hy2z  +  Iyz2  +  Jxyz  =  0, 

where  A,  B ,  C,  •  •  •,  are  arbitrary  constants  and  x,  y,  z  are  homo- 
geneous coordinates,  be  the  general  equation  of  a  cubic. 

If  the  cubic,  /=  0,  is  to  pass  through  the  six  points  P0  the  coef- 
ficients A ,  B ,  •  •  • ,  must  satisfy  the  six  equations 

(2)  Ax3  +  By3  +  Ck\  +  •  •  •  =  0        (*  =  i,  2,  •  -,  6). 

The  matrix  of  the  coefficients, 

m=\\x\      y3      z\      x2Vi      x2z.      x.y2      xz\      y2z.      y.z2     x.ytz.\\y 

is  always  of  rank  six ;  that  is,  at  least,  one  of  its  determinants  of 
order  six  is  4=  0 .  For,  if  we  choose  the  vertices  of  the  triangle  of 
reference  to  coincide  with  the  points  P19  P2,  P3,  respectively,  we 
have 

and  therefore,  since  no  three  of  the  six  points  are  to  lie  on  a  straight 

6 


PROPERTIES   OF   LINEAR   SYSTEM. 


line,  xv 
straight  line, 


,  z6  =}=  0 ;  moreover,  since  P4 ,  P6 ,  P6  do  not   lie  on  a 


*4 

Vi 

zt 

X5 

ft 

Z5 

X6 

ft 

\ 

+  0. 


Therefore  the  determinant 


%Va    xiVl    x4Vih 

xly5  x*yl   x*v*h 
xly6  %y\  *&** 


to  which,  in  the  present  case,  one  of  the  determinants  of  9ft  reduces, 
is  =f  ®y  and  consequently  the  rank  of  9ft,  which  is  invariant  under  a 
transformation  of  coordinates,  is  six.  At  the  same  time,  we  obtain 
from  ( 2  ) ,  for  this  special  triangle  of  reference,  A  —  B  —  (7=0; 
and  if  we  solve  equations 


(3) 


Dx\y.  +  Ex\z.  +  Fx.y\  +  •  •  •  =  0  (t  =  4,  5,  6), 


with  respect  to  D,  F,  J  and  substitute  the  values  in  (1),  we  get, 
for  the  system  of  cubics, 

JI(a;^  +  fi[xy  +  y[xz  +  yz)y  +  G(a'2xy  +  fi'2tf 
(4)  +  7>  +  *)*  +  J«<*  +  P'3xy  +  Vsxz  +  z*)y 

+  E{a[xy  -f  fi\f  +  7^  +  **)*  =  °; 
where  a[ ,  £J ,  •  •  • ,  are  fixed  constants.     Hence,  if  we  denote 
fi  s  K^  +  #^2/  +  7^2  +  yz)y  s  ^y, 

Si  =  K*2/  +  ^y2  +  y'2yz  +  **)*  ■  <M> 

/3  =  K*2  +  £3  xy  +  7>2  -f  22)2,  =  <j>3y, 
/4  =  (a^zy  +  /S^i/2  +  7^2;  +  xz)x  =  <£4x, 

a  =  H,  fi=  G ,  7  =  /,  B=z  E;  we  obtain  the  totality  of  cubics 


(5) 


8  SINGULARITIES   OF  ALGEBRAIC  CURVES. 

through  the  six  points  in  the  form 

(6)  2«*/i+>/J+')'/,  +  */«-0, 

a,  /3,  7,  8  being  arbitrary  constants.  The  four  cubics/^,  ft9  fi9  f4 
are  linearly  independent ;  that  is,  it  is  impossible  to  determine  four 
constants  a,  ft,  7,  8,  not  all  zero,  so  that 

(7)  a/1  +  /3/2  +  7/3+8/>0, 

since  each  one  of  these  four  cubics  f19  f2i  fs,  jf\  contains  a  term 
which  does  not  occur  in  the  other  three ;  namely,  the  terms  y2z ,  xz2 , 
yz2 ,  x2z  respectively.  Hence  the  cubics  through  six  points  form,  with- 
out exception,  a  triply  infinite  system,  provided  that  no  three 
of  the  six  points  lie  on  a  straight  line. 

§  2.   Eeduction  of  the  base  cubics  to  canonical  form. 

Instead  of  the  four  base  cubics  flt  /2,  fz>  fA,  we  introduce  four 
other  base  cubics  each  of  which  breaks  up  into  three  straight  lines, 
in  the  following  manner :  fx  and  fs  have  the  common  factor  y,  and 
the  two  conies,  $t  =  0,  <\>z  =  0,  pass  through  the  four  fundamental 
points  P2,  P4,  P5,  P6;  hence  the  pencil,  afa  +  y<f>3  =0 ,  contains 
the  product  of  the  two  lines,  P±P6,  P^P** >  say  f°r  a  =  «',  7  =  7', 
and  also  the  product  of  the  lines,  P4P5,  P2P&>  say  ^or  <*  =  a"> 
7  =  7".     Hence  the  two  cubics, 

break  up  into  the  lines  P1 P3  P^-  P2 P5  and  PXPZ  P4P5-  P2 P6  re- 
spectively. Similarly  the  constants  /3',  8'  and  /3",  8"  can  be  so 
selected  that  the  two  cubics, 

f  £-*/,+>/« 

break  up  into  the  lines  P2  P3  •  P4  P6  •  P,  P6  and  P2  P3  •  P,  P,  •  P4  P5  re- 
spectively. 


PROPERTIES    OF    LINEAR   SYSTEM. 


Since  a  <y"  —  y'a"  and  /3'8"  —  ft"  8'  are  evidently  =)=  0,  the  four  new 
cubics,  fx ,  f2,f3,f4,  are  again  independent  and  can  therefore  be  used 
for  base  cubics. 

In  order  to  obtain  the  expressions  of  the  cubics,  J1}  f2)  fz,  f4,  in 
terms  of  x,  y,  z,  let  the  equations  of  the  lines,  P5P6,  P4P6,  -^4^5?  be 
respectively, 

r  \  ==  aYx  -f  a2y  +  azz  =  0, 

(10)  J  tissb1x  +  b2y  +  b3z=0, 

v  =  cYx  -f  c2y  4-  c3z  =  0, 

where  a19  •  •  • ,  c3  are  constants. 

Since  the  points,  P4,  P5,  P6,  are  distinct  and  no  three  of  the 
points,  Pi  (i  =  1 ,  2,  •  •  •,  6),  shall  lie  on  the  same  straight  line,  the 
coefficients,  aiy  •  •• ,  c3,  must  satisfy  the  following  conditions  : 


i) 


A  = 


al 

a2 

as 

h 

K 

h 

Gl 

G2 

C3 

+  0; 


2)  all  the  first  minors  of  A  =)=  0,  since 

6ic3:6ic2  — ^2cu  etc-; 
63c2  =  0,  the  points,  P2,  P3,  P4,  would  lie  on 


xi:yi:zi^b2c3-bic2:bicl 


if,  for  instance,  &2c3 
the  same  line,  etc.; 

3)  all  the  elements  of  A  =(=  0 ,  for  if,  for  instance,  a,  =  0 ,  the  line 
P5P6  would  pass  through  the  point  Px ,  etc. 

Calling  the  new  base  cubics  again  flf  f2,  f3,  f4,  their  analytic  ex- 
pressions are 


/lmy4>lmw(&-<V)mw(eiz-l3f>)mPtPt-PlPt-PtPv 
ftmx^tm^(e1\-a1p)m^(fia-^)mPtPt  ■  P2P3  ■  PXP„ 


en)  \J^n r 

K     ;  \ftma*tm 
(12)   l^"^" 


w(6IX-o1/.)«,*(W!_7,y)«P,P,.P1P,.PIP,, 


where  a1}  ■  •  • ,  %  are  the  minors  of  A  corresponding  to  the  elements 
«x ,  •  •  • ,  c3  respectively.  From  the  preceding  normal  form,  the  fol- 
lowing results  are  easily  proved  : 


10 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


1)  The  base  cubics  have  no  other  common  point  of  intersection 
except  the  six  points  P.  (  i  —  1 ,  2 ,  •  •  • ,  6  ) . 

2)  Not  one  of  the  points,  Pif  is  a  multiple  point  on  all  the  base 
cubics. 

3)  At  none  of  the  points,  Pif  have  all  the  base  cubics  coinciding 
tangents. 

(Compare  the  adjoining  diagram.) 

§  3.   Eank  of  the  matrix  of  the  first  derivatives  of 
We  consider,  in  this  section,  the  matrix 

J\x         Jlx         J3x         Jix 

\A")  Jly        Jly        JZy         J  iy 

J\z         J-2z  J$z  Jiz 


where 
(14) 


Jix  —   dx    > 


df- 

f.  —  — 

J  iy  Qy  > 


Jiz  -      dz    > 


and  propose  to  determine  all  points  in  the  plane  for  which  the  rank, 
r,  of  the  matrix  is  0 ,  1 ,  2  respectively. 

1°  r=0. 

When  r  =  0 ,  all  the  elements  in  the  matrix  vanish ;  therefore, 
since 

Xfix  +  yfiy  +  Zfi,=   Zfi> 

the  point  is  a  common  point  of  intersection  and  moreover  a  double 
point  on  each  base  cubic,  against  the  result  already  established. 
Therefore,  there  are  no  points  in  the  plane  for  which  r  =  0. 

2°  r=l. 

When  r  —  1 ,  all  the  determinants  of  order  two  vanish  ;  whereas,  at 

least,  one  element  is   =f=  0.     Hence  it  follows   that  at  a  point   in 

which  r  =  1 ,  the  four  base  cubics  have  a  common  tangent,  since,  at 

such  a  point, 

f  .  f   .  f   _.  f   .  f   .  f  . 

'  ix  *  J  iy  * «/  iz         J  jx  *  J  jy  *   'jz  f 


PROPERTIES   OF   LINEAR   SYSTEM. 


11 


combining  this  result  with  the  remark  at  the  end  of  §  2 ,  we  see  that, 
at  a  fundamental  point,  the  rank  r  can  never  be  =  1 .  The  points, 
where  r  =  1 ,  are  found  by  equating  to  zero  all  determinants  of  order 
two. 

Consider  first  the  determinants 

(***)       fix  JZz  ~J\z  Jte  ~  ^  f       fix  Jzy         J\y  J Sx  =  ^  >       J  \y  J 3*  """./ Is  J Zy  =  "  * 

Therefore 


(16) 


where 


a** 


*,■ 


^ 


<A.  = 

r  .2 


^*, 


From  equations  (16),  we  have  either  y  =  0,  or  simultaneously, 

(17)  KK-KK  =  ®,  *»K-+*+i,-0'  ***•.- M*-0 

The   points,  which  satisfy  these  equations,  are   the   double  points, 
©j,  a>2,  ©3,  of  the  pencil  of  conies 


(18) 


*4>i  +  7tf>3  =  °- 


The  conies,  <^1  =  0,  <£3  =  0,  pass  through  the  points,  P2,  P4, 
P5,  P6;  and  the  double  points,  colf  <w2,  g>3,  of  the  pencil,  are  the 
three  diagonal  points  of  the  complete  quadrangle  P2PAP5P6. 

Consider  also  the  determinants 

VA*V      J2yJiz        JZzJiy  ==  ^  f  J2yJix        J2xJiy  =  "  >  J 2xJ 4z        JlzJix  "*  ^  * 

Therefore 


(20) 


12  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

From  these  equations,  we  have  either  x  =  0,  or  simultaneously, 

(2i)  M* - K+*  =  °> K**> -*A-°. **** - KK  =  °; 

and  the  points,  which  satisfy  these  equations,  are  the  three  diagonal 
points,  ft)j ,  a>2 ,  «3 ,  of  the  complete  quadrangle  Pj  P4  P5  P6. 
It  is  easily  proved  : 

1)  The  points,  ml3  co2,  g>3,  do  not  lie  on  the  line,  x  =  0. 

For,  if  we  suppose  one  of  the  points,  say  o>1 ,  the  intersection  of 
the  lines,  P2  P6 ,  P4P5,  to  lie  on  the  line  x  =  0 ,  which  passes  through 
the  points,  P2,  P3,  then  the  two  lines  P2P3  and  P2P6  would  contain 
the  same  point  <ol  (  different  from  P2 )  and  would  coincide ;  hence, 
the  three  points,  P2,  P3,  P6,  would  lie  on  the  same  straight  line, 
against  the  hypothesis.  In  like  manner,  it  follows  that  the  points, 
a>2 ,  g>3  ,  do  not  lie  on  the  line,  x  =  0 . 

2)  Similarly,  the  points,  <o[ ,  a>2 ,  g>3  ,  do  not  lie  on  the  line,  y  =  0. 

3)  None  of  the  points,  a>lt  a>2>  g>3,  coincides  with  any  of  the 
points,  •!,«£,  »;. 

If  we  suppose  the  point,  c^,  to  coincide  with  any  one  of  the 
points,  say  (d[,  then  the  two  lines,  P2P6,  P5P&,  would  contain  the 
same  point,  a>J,  and  would  coincide ;  hence,  the  three  points,  P2,  P5, 
P6,  would  lie  on  the  same  straight  line.  In  like  manner,  it  follows 
that  the  point,  (al ,  does  not  coincide  with  either  <a'2 ,  or  g>3  .  Simi- 
larly, we  find  that  neither  one  of  the  points,  o>2,  o>3,  coincides  with 
any  of  the  points,  (o[ ,  ©2 ,  a>3 . 

4)  The  lines,  x  =  0,  y  =  0,  meet  at  the  point,  P3 ;  the  point,  P3, 
is  therefore  the  only  point  in  which  the  six  determinants,  (15)  and 
(19),  vanish.  But  we  have  already  shown  that,  at  a  fundamental 
point,  not  all  the  determinants  of  order  two  can  vanish  ;  therefore, 
there  are  no  points  in  the  plane  in  which  r  =  1 . 

3°  r  =  2. 

When  r  =  2 ,  all  the  determinants  of  order  three  vanish  ;  whereas, 
at  least,  one  determinant  of  order  two  is  =4=  0 . 

Such  points  are  the  common  points  of  intersection  of  the  Jacobians 
of  the  four  nets  of  cubics, 

(22)  aA  +  /3/2  +  7/3  =  0 ,         a'f,  +  Pft  +  B'ft  =  0 , 


PROPERTIES    OF    LINEAR   SYSTEM. 

(23)    </;  +  7y3  +  ay,  =  o,      /r/2  +  yy  +  «""/<  =  o . 

Consider  the  Jacobian, 


13 


(24) 


',- 


J\x  J\y  Jlz 
Jix  JZy  JZz 
Jix         J  Ay         Jlz 

of  the  net  of  cubics, 

(25)  <£+■#; +*y,-o 

By  an  easy  reduction,  we  have 

</>!*  <t>ly  <f>\z 
♦*  <l>3y  $3* 
4>4x        <t>*y        <t>4z 


yK 

*1  +  #iv 

yK 

vK 

4>3  +  ytly 

y+* 

<t>*  +  #., 

xK 

^4. 

(26)       J-2  =  |y  Jay 


+  2<#>4 


#1.       <k» 


=  0, 


and  finally, 


(27) 


If  we  interchange  fi  and  i>,  6  and  c,  in  ( 11 )  and  ( 12 ) ,  fx ,  /2,  /3, 
/4  undergo  the  substitution  {fifz){J2fi)-  Therefore,  we  obtain  for 
the  Jacobian  J4, 

J  Sx  J 3y  J  Zz 
J\x  J\y  JU 
J2x        Jly        Jlz 


(28) 


Ji? 


(29) 


+  a2Aa5/*v(c1X  —  a^)  —  b2fi3Axy\v']  =  0 


The  Jacobians,  J2,  J4J  have  the  line,  y  —  0 ,  as  a  common  factor 
and  intersections  at  the  points,  P17  P2,  P3,  P4,  P5,  P6,  a^,  ©2,  a>3, 
two  further  points,  (^ ,  Q2 ,  and  only  at  these  points.     For,  if  we  set 


(30) 


J,  =  -  3yf  4  =  -  3?/*%  =  °, 


(31) 


14  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

where 

*Xa=(61X~alM)(c2X-a2^)(62X-a2/x)(c3/i~6sz/) 

-|-  ^AxfMv^X  —  axn)  -f  c2y3Axy\fi 

+  a2Axfiv(y2z  -  7sy)  -f  ct%JSxtfkfi, 

-f  ^Ax/jlv^X  —  axv)  —  b2fi^AxyXv 

■  (£32/  -  P2z)(Piz  ~  Psx)(%v  -  71^)(«12/  -  «»«) 

4-  a2Axfiv(P5y  -  $2z)  -  b2PsAxyXv, 


(32) 


then  the  points  of  intersection  of  -^2  =  0  and  ^r4  =  0  can  be  deter- 
mined as  follows : 

1  °  /i  =  0  and  v  =  0  intersect  at  the  point  P4 . 

2°  /i  =  0  and  %2  =  0  intersect  at  the  points,  P4,  g>3,  and  P6,  P6, 
since  the  point  P6  is  a  double  point  on  %2  =  0 . 

3°  v  =  0  and  %4  =  0  intersect  at  the  points,  P4,  ©j,  and  P5,  P6, 
since  the  point  P5  is  a  double  point  on  ^4  =  0 . 

4°  %2  =  0  aud  %4  =  0  intersect  at  the  points,  Plf  P2,  P3,  P4,  P5, 
P6 ,  o)2 ,  and  at  the  two  points,  Q19  Q2,  determined  by  the  equations, 


(33) 


Each  of  the  points,  P5  and  P6 ,  counts  for,  at  least,  two  points  of 
intersection,  since  P5  is  a  double  point  on  %4  =  0 ,  and  P6  is  a  double 
point  on^2=0. 

The  point  P2  is  a  double  point  on  both  %2  =  0  and  ^4  =  0,  and 
counts  therefore,  at  least,  for  four  points  of  intersection.  At  P3,  the 
curves,  %2  =  0 ,  %k  —  0 ,  have  coinciding  tangents,  whose  equation  is 

(34)  x(a2Abzcs  +  fl^x)  -  S^ft1?!  =  °- 

At  jP1 ,  the  equations  of  the   tangents  to   %k  —  0  >  X2  =  0   are> 


\c 


PROPERTIES   OP   LINEAR   SYSTEM. 

respectively, 


(35) 


Aa, 
&„y ( 1  —  ob~c, )  —  #,3  =  0,  where  o  ==  ~ — — r — jr 


twO  -^ic2)-,y22  =  o, 

and  because 


(36) 


7,(1 -Aci)      7 


=  0, 


the  two  tangents  coincide.  Hence  each  of  the  points,  P3  and  P1 , 
counts,  at  least,  for  two  points  of  intersection. 

Counting  each  point  of  intersection  with  the  minimum  multiplicity 
thus  determined,  we  obtain  a  total  of  sixteen  points  of  intersection,  the 
exact  number  of  'points  of  intersection  of  the  two  biquadratics,  %2  —  0  ? 
%4  =  0.  Hence,  the  two  biquadratics,  ^2  =  0 ,  j£  =  0 ,  can  have 
no  other  points  of  intersection  than  those  enumerated  above ;  and 
the  Jacobians,  J2,  J4,  have  the  line,  y  =  0 ,  as  a  common  factor,  and 
intersections  at  the  points,  Pl3  P2,  P3,  P4,  Pb,  P6,  coiy  a>2,  a>3,  Ql , 
Q2 ,  and  only  at  these  points. 

If  we  interchange  x  and  y ,  subscripts  1  and  2 ,  in  (11)  and 
( 1 2  ) ,  fx ,  /s, fM  ,f4  undergo  the  substitution  (f  f2 )  (/3  f4 ) .  There- 
fore we  obtain,  for  the  Jacobians,  Jx ,  J3, 

J1  =  3xv[(b2\  —  a2u)(blX  —  a1p)(cl\  —  a1v)(eifi  —  b^v) 
I     '  —alAyuv(b2\  —  a2u)  +  cly3Axy\p]=0, 

/3=  _3a;/i[(c2X-a2i/)(61\-a1/x)(c1X-a1i/)(c3/x-63z/) 
^      '  —  alAyuv(c2\  —  a2v)  —  b^/S^kxyXv]  =  0. 

The  Jacobians,  J3 ,  JJ, ,  have  the  line,  x  =  0 ,  as  a  common  factor, 
and  intersections  at  the  points,  Pl9  P2,  P3,  P4,  P6,  P6,  w[,(o2, 
o>3 ,  Q[ ,  Q2,  and  only  at  these  points  ;  Q[ ,  Q2 ,  are  on  x  ==  0 ;  and 
since  none  of  the  points,  (*>[,  co'2,  o>3 ,  Q[ ,  Q2 ,  coincides  with  any  one 
of  the  points,  mls  a>2,  ©3,  Ql9  Q2,  we  have  the  lemma : 

There  are  six,  and  only  six,  points  in  the  plane  II  in  which  r  =  2  ; 
namely,  the  six  fundamental  points,  Pi ,  and  r  =  3 ,  AT  all  other 

POINTS. 


CHAPTER   II. 

The  One-to-One  Correspondence  Between  the  Cubic  Sur- 
face and  the  Plane. 

§  4.   Transformation  from  plane  to  cubic  surface. 

As  in  Chapter  1 ,  let  Pv  ■  •  • ,  P6  be  six  distinct  points  in  a  plane 
II ,  no  three  of  which  lie  on  a  straight  line,  and  further  let 

(1)  /,(*.*»•  *)**0,        f,(*,l*,v)-0, 

where  X,  /a,  i>  denote  any  system  of  trilinear  coordinates,  be  the 
four  base  cubics  of  the  triply  infinite  system  passing  through  the  six 
points. 

Then,  if  x1  :x2:xs:xA  denote  tetrahedral  coordinates  of  a  point  in 
space,  the  equations, 

(2)  m—/u         ^2=/2>         (*****/*>         P*A=f» 

define,  in  parameter  representation,  a  cubic  surface  i^3  and,  at  the 
same  time,  a  correspondence  *  between  the  points  of  the  plane  and 
the  points  of  the  cubic  surface. 

a)  To  any  point,  P0 ,  in  the  plane  II ,  different  from  the  points, 
jP1?-  •  •,  JPr,  corresponds  one  point,  P'Q ,  on  the  surface  i^3;  since,  not 
all  four  of  the  quantities,  f\ ,  f\ ,  f\ ,  f\ ,  equal  zero,  and  the  ratios, 

x1 :  x2 :  x3 :  a?4  =/ 1  :j2 : / 3 :/  J , 

have  definite  values  ',  f\,  f\,  f\,  fl  are  the  values  of  the  functions, 
fufzi  fs>  ft>  respectively,  at  the  point  P0.  The  six  points,  Plt 
•  ••,  P6,  for  which  the  functions,  flf  f2,  fif  f4)  vanish  simulta- 
neously, are  called  fundamental  points. 

b)  To  a  fundamental  point,  together  with  the  totality  of  the  paths 
of  approach  to  it,  corresponds  a  fundamental  curve  on  P3. 

*  Clebsch,  "  Geometrie  auf  den  Flachen  dritter  Ordnung,"  Crelle,  65. 

16 


ONE-TO-ONE   CORRESPONDENCE.  17 

Let 

(3)  \  =  \  +  \t}         j£  =  A*0+V>         v=vo+vit> 


(4) 


14=0/ 


be  a  path  of  approach,  L,  to  a  fundamental  point,  P0.     At  P0, 

/J  -/!-/{ -/i.-o, 

but  according  to  §  3,  not  all  of  the  quantities,/^,  •  •  •,  f°4v}  =  0; 

Expanding  the  functions,^,  y^,/^,/^,  in  the  vicinity  of  P0,  we  get 

(5)  ^•=/?x(^\)+/?m(^-^o)+/>(^-^)+---)  0  =  1,2,3,4), 
and  by  (3)  .  • . 

(6)  <ra,  =  X^  +  ixJl  +  *>/•  +  powers  of «,         <r  -  t . 

If  we  approach  to  the  limit,  £=0,  the  ratios,  xl:x2:xi:xA>  ap- 
proach to  definite  values, 

(7)  '«*-\fk  +  *Jli'+*Jl> 

and  define  a  definite  point.     For,  suppose 

(7«)    \Jl  +  ^fl  +  ^n  =  0,         for        /- 1,2,3,4. 

Then,  since  by  §  3  at  a  fundamental  point  the  rank  of  the  matrix 
of  the  coefficients  is  two,  these  equations  define  a  unique  solution  for 
\lifi,l:vl;  and,  since 

V7»  +  Mo/?*  +  "of%  =  3/J  =  o, 
it  follows,  that  \  :  fiQ :  j/q  is  a  solution  for  the  same  equations ;  hence, 

\L  : /V  ^  =  \0 : /VV 

against  the  assumption  (4). 

*By  this  notation,  we  mean  that,  at  least,  one  of  the  determinants,  \fix — Pq^  , 
\vi  —  vt\ » /"ovi  —  vo*"i >  is  different  from  zero. 


18 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


c)  Two  different  lines  of  approach  to  P0  determine  two  different 
points  on  the  fundamental  curve. 
Let 


i    Mi 


+  o, 


be  another  path  of  approach,  L' ,  to  P0,  not  coinciding  with  L,  that  is 


(9) 


\ 

K 

vo 

\ 

i*i 

Vj 

K 

K 

k 

+  0. 


If  the  two  points,  P^,  P^',  on  the  fundamental  curve  correspond- 
ing to  the  lines  Z,  L' ,  coincide,  then 

(io)    p(\fl  +  vji  +  vji)  =  x;/jA  +  /,;/^  +  v[fi 

(i  =  l,2,  3,4). 

Therefore, 

(i i)    (,\  -  x;  )/j»  +  (m  -  k  )fu  +  (p"i  -  K >/',  =  o • 

By  §  3,  the  rank  of  the  matrix  of  the  coefficients  in(ll)isr=2, 
and,  by  the  Euler  equations,  we  know  \:/*0:  v0  is  a  solution  of  these 
equations  ;  hence,  it  follows,  that 


(12) 


\ 

n 

vo 

\ 

Pi 

"l 

K 

K 

r 

=  0 


against  the  assumption  ( 9  ). 

The  point,  P^ ,  describes  the  fundamental  curve  as  the  parameters, 
\  >  Pi  9  vi9  ta^e  a^  possible  values. 

d)  A  fundamental  curve  on  Fz  is  a  straight  line. 

For,  according  to  §  3,  the  rank  of  the  matrix, 


(13) 


\\J  jk       JjfJ.       Jjl 


(.7  =  1,2,3,4), 


ONE-TO-ONE  CORRESPONDENCE.  19 

is  r=2.  Hence,  if  we  suppose,  for  instance,  that/JK/^—  /5M./,JV  +  0  > 
two  factors  m'  and  »'  can  be  found,  so  that, 

.«  =  «#£  +  »:#; 

and  these  values  substituted  in  (7)  give 
(14)  axj  -/£  (m'\  +  /*,)  +</5J(VXI  +  »,)• 

If  we  set  m'X  +  /i,  =  <j ,  »'X,  +  »,  =  t2,  we  have 

(is)  ««v^:<,+^Vi 

the  equations  of  a  straight  line  in  parameter  representation.  There 
are,  therefore,  six  fundamental  lines  on  F3  corresponding  to  the  six 
fundamental  points,  PIS  •  •  •,  P6,  in  the  plane,  II. 

§  5.    The  one-to-one   character    of    the   correspondence 
between  the  plane  and  the  cubic  surface. 

a)  To  two  non-fundamental  points,  in  the  plane  II,  always  corre- 
spond two  different  points  on  F3,  provided  the  additional  condition  be 
imposed  upon  the  fundamental  points ,  that  the  six  points,  Piy  •  •  •,  P6> 
do  not  lie  on  a  conic. 

If  to  a  non-fundamental  point,  P0,  corresponds  the  same  point  on 
F3  as  to  the  non-fundamental  point,  P ,  then 


(16) 


A   A   A   A 

J  1         «/ 2        J  3        J  i 


=  0. 


The  quantities,  f\ ,  f\ ,  f\ ,  f\ ,  are  not  all  equal  to  zero  ;  suppose 
f\  =(=  0 ;  (1  6  )  is  then  equivalent  *  to  the  three  equations, 

(17)   AA  ~AA  =  o ,  Afi  -AA  =  o ,  AA  -Af  -  o . 

These  cubics  have  either  a  common  factor  or  no  common  factor. 
1  °  A  common  factor  must  be  of  degree,  n  =  1 ,  2 ,  3 . 

1)  n=l. 

Let  the  greatest  common  factor  L  of  the  three  cubics  be  of  degree 
*Kkoneckeb,  "  Bemerknngen  zur  Defcerminanten-Theorie,"  CreUe  72,  p.  152. 


20  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

n  =  1,  in  X,  /*,  v)  then,  by  (17),  we  have 


(18) 


JWA        /4/1  "  ^'Xzi 


where  %x ,  %2 ,  X3  are  quadratic  factors. 

Since  no  three  fundamental  points,  Px ,  •  •  • ,  P6 ,  lie  on  a  straight 
line,  then,  if  two  of  them  lie  on  L  —  0 ,  must  the  remaining  four  lie 
on  each  conic,  %J  =  0,  x2  —  0,  %^  =  0 ;  because  each  cubic  in  ( 18  ) 
passes  through  each  fundamental  point.  Therefore  %x ,  %2  >  X3  belong 
to  the  same  pencil ;  but  this  would  involve  a  linear  relation  between 
fufzyfsift)  which  is  against  our  hypothesis. 

If  only  one  fundamental  point  should  lie  on  L  —  0 ,  then  the  re- 
maining five  fundamental  points  must  all  lie  on  each  conic  x[  >  X2  > 
Xs ;  hence  %[  >  Xt  t  Xs  would  coincide,  and  therefore  the  common 
factor  in  (18)  would  be  of  higher  degree  than  w=  1,  against  the 
assumption. 

If  no  fundamental  point  should  lie  on  the  line  L  =  0 ,  then  all  six 
fundamental  points  must  lie  on  each  conic,  x[ sm  fy  X2  =  0  >  Xi  ess  ^1 
against  the  hypothesis. 

2)  n  — 2. 

Let  the  greatest  common  factor  i/r  of  the  three  cubics  be  of  degree 
n  =  2,  in  X,  /a,  v ;  then,  from  (17),  we  have 


(19) 


Afi-Afi-ir-h, 
Af>-Afx**k> 

\-Aft-AA-+-h> 


where  l19  l2>  ls  are  linear  factors.  Since  all  six  points,  Plt  •  •  •,  P6, 
do  not  lie  on  a  conic,  then,  if  five  of  them  should  lie  on  ty  =  0 ,  the 
remaining  one  must  lie  on  each  line,  lx  =  0 ,  l2  =  0 ,  ?8  =  0 ;  hence  a 
linear  relation  must  exist  between  llf  l2,  l3,  and  therefore  between 
fiyfzyfzifv  against  the  hypothesis.  If  less  than  five  fundamental 
points  should  lie  on  yjr  =  0,  then  the  remaining  ones  must  lie  on 
each  line,  lx  =  0 ,  l2  =  0 ,  /3  =  0  ;  hence  these  lines  would  coincide, 


ONE-TO-ONE    CORRESPONDENCE. 


21 


and  therefore  the  common  factor,  in  ( 19  ) ,  would  be  of  degree  n  —  3 , 
against  the  assumption. 

In  like  manner,  it  is  easily  proved,  if  n  =  3 ,  that  a  linear  relation 
exists  between  f19  f2,  f3,  f4,  in  which  not  all  the  coefficients  equal 
zero.  Therefore,  it  follows,  that  the  three  cubics  ( 17  )  do  not  contain 
any  common  factor. 

2°  Since  the  three  cubics,  in  (17),  do  not  contain  any  common 
factor,  they  can  intersect  in  only  a  finite  number  of  points  ;  we  know 
already  seven  points  of  intersection,  P0>  Pl9  •  •  • ,  P6 . 

Suppose  one  additional  point  P ,  different  from  the  seven  points, 
P0,  jPj,  •  •  •,  P6,  to  be  a  point  of  intersection  of  these  cubics,  then 
they  meet  in  eight  points  and  therefore  have  a  ninth  base  point  in 
common ;  hence,  they  belong  to  the  same  pencil  and  one  of  them  is 
expressible  linearly  in  terms  of  the  other  two. 

Set 

(20)  P2(flf2  -fifi)  +  p3(/J/s  -/S/J  +  P^Af-flA)  =  0, 
where  p2)  p3,  p4  are  not  all  equal  to  zero;  therefore 

(2i)  -/, (ft/;  +  pji  +  ft/n + ft/?/, + ft/j/3 + ft/?/4 = o, 

since  p2,  p3,  pi  are  not  all  equal  to  zero  and  f\  =f=  0,  the  coefficients 
of  this  linear  relation  can  not  all  be  zero ;  but  this  is  a  contradiction 
to  the  previous  result  that /J,  f2,  f%,  f±  are  linearly  independent. 

b)  The  image  of  a  non-fundamental  point  does  not  lie  on  a  fun- 
damental line. 

For,  if  the  image  of  the  non-fundamental  point,  P,  coincides 
with  the  point  on  the  fundamental  line  corresponding  to  a  funda- 
mental point,  P0(\,  p0>  v0),  determined  by  the  linej*  of  approach, 


(22) 


\=\  +  \t,     11  =  ^+^1,     v^v^+vj], 


then 
(23) 

\     H     "o| 

/ 1     J  2     Jz 

S1    82     &3 

*< 
84 

*  Compare  \  4,  6. 

=  0, 


22  SINGULARITIES   OF   ALGEBRAIC  CURVES. 

where 

(24)  Sj  =  \fl  +  Kfl  +  "if I  U  =  h  2,  3,  4)  ; 

not  all  of  the  quantities  Slf  S2,  Ss,  84  are  equal  to  zero. 

Suppose  /Sj  4=  0 ;  then  (23)  *  is  equivalent  to  the  three  equations, 

(25)  jfa  -fA  =  0,  /tSs  -f.S,  =  0,  fX  CfA  =  °- 

It  follows  then  exactly  as  under  a)  that  these  three  cubics  can  have 
no  common  factor ;  they  intersect  therefore  in  a  finite  number  of 
points,  and  we  know  already  seven  points  of  intersection  ;  namely, 
the  six  fundamental  points,  one  of  which,  _P0,  counts  twice,  since, 
at  the  point  P0,  the  line  (22)  meets  each  cubic  in  two  coinciding 
points;  for,  if  we  substitute  \,  fi,  i>  of  (22)  in  (25)  and  expand 
according  to  powers  of  t ,  the  expansions  begin  with  the  second  power 
of  t.  Hence  we  infer  as  under  a)  that  the  three  cubics  (25)  can 
have  no  further  point  in  common. 

c)  The  image  of  a  non-fundamental  point  is  not  a  singular  point  on 
the  surface  F3. 

Let  P0  be  any  non-fundamental  point  and  consider  the  line  L 
through  P0, 

(26)  \  =  \  +  \t,  \i  =  fiQ  +  f^t,  P**  v0+  Ptt9 


(27) 


*0 


II  X0       ^o       vo 

Ik  ft  vi 

The  image  P'9  of  P0  is  given  by 

(28)  xl:x2:x3:xi=fl--fl-fl--A, 
and  the  tangent  to  the  image  of  L  at  P  'Q  by 

(29)  x,  =f\  +  V,  x2  =/»  +  82t,  x,  =/»  +  Bfo  xt  =fl  +  SJ. 
By  the  Euler  equations,  we  get 

(30)  3/;  =  \„/°A  +■*/£  +  "of%  0  =  1,2,3,4) 
and  substituting  (24)  and  (30)  in  (29),  we  get 

(31)  P"xj=/U\  +  3V)  f&fc  +  3V)  +fK"o  +  3"i0- 
*Kronecker,  OreZte,  72. 


ONE-TO-ONE   CORRESPONDENCE. 


23 


Set  X0  -J-  3X^  =  X',  a*0  +  3a*!  t  =  a*' ,  vQ  +  3^  =  v  ,  therefore 
(32)  Ay  =  ^  +  ^  +  ^J. 

Since,  according  to  §3,  the  rank  of  the  matrix,  \\f%f%  fQjV\\i  *s 
r  =  3 ,  the  equations  (  32  )  represent,  in  parameter  representation  with 
the  parameters  X',  a*',  v  ,  a  fixed  plane  through  P0',  independent  of 
Xj,  a*i,  vl9  and  the  tangent  (29)  lies  in  this  plane.  The  surface  F3 
has  therefore  atPJa  determinate  tangent  plane  and  the  point  P'0  is 
a  non -singular  point. 

d)  Two  fundamental  lines  on  the  surface  F3  have  no  common  point. 

If  the  fundamental  line  L[  corresponding  to  Px  and  the  funda- 
mental line  Z3  corresponding  to  P3  have  a  common  point  P'  deter- 
mined on  L[  by  the  path  of  approach  to  P1  ( 1 ,  0 ,  0  ) ,  (where  X,  a*,  v 
are  trilinear  coordinates  with  respect  to  the  triangle  of  reference 
PlP2Pi)  these  three  fundamental  points  are  its  vertices) 


(33) 


x  =  i+x;*,  ^  =  o+a*;*,  p=o+vit9 

10      0 


K  f*[  v[ 


+  o, 


and  on  L'5  by  the  path  of  approach  to  P3  (  0 ,  0,  1 ) , 


(34) 

then 
(35) 
where 

(35.) 


\  =  o+v,    /t  =  o+/«1<, 

0     0      1 


1+Vlt, 


+  o, 


Sf    Sf    #£>    Sfl 

«<»    -Sw    Sg>    fl»|  =  °'  ' 

f»„(dJi\     ...  f<»={?£\ 


24  SINGULARITIES  OF   ALGEBRAIC  CURVES. 

By  the  canonical  form  of  f19  f2,  f3,  f4,  we  find, 


(35,) 


I  ^(3l)  -  ^;ci73>         £?>  -  -Kci,y3+^172. 


Since   not  both  \ ,  /^  equal   to   zero,  suppose   pt  4=  0 ,  therefore 
>S'1(3)  =j=  0  ;  then  (35)  is  equivalent  to  the  three  equations, 

(36)    S^Sf-SfS^O,    8p8®-8p8n**09    Sf S?-S*> Ajf>-0. 

Therefore, 

(^ftft  -  \PA)ti  -  Pgfiifal  =  0,  I ; 

(37)    .  (ta/ta-tatoiMr*!     n; 

.(\^^7rAMA*)K  +  ^ft^i"''     IIL 

In  I  and  III,  if  ^  =  0 ,  then  ^  =  0 ,  since  /^  4=  0 ;  against  the 
assumption  in  (33) ;  therefore,  we  have  fi[  4=  0. 
In  II,  since  fi[  4=  0,  we  have 


(38) 


&A/*i*t-^«bft7i" 


dx  6X     a2  Cj     6X  c1 

«262        a2C2        62C2 
«363        a3C3        63C3 


=  0, 


the  necessary  and  sufficient  condition  for  all  six  fundamental  points 
to  lie  on  a  conic,  against  the  hypothesis.  In  like  manner,  it  follows, 
if  fil  =  0>  that  all  six  fundamental  points  lie  on  a  conic.  What  has 
just  been  proved  for  the  two  fundamental  points  Pt ,  P3  holds  for  any 
two  fundamental  points.  Hence  two  fundamental  lines  on  the  surface 
F3  have  no  common  point 

As  a  result  of  these  theorems,  it  follows,  that  to  every  point  of  F3, 
not  on  a  fundamental  line,  corresponds  one,  and  but  one,  point  in  II, 
without  exception  ;  and  to  every  point  of  Fz,  on  a  fundamental 
line,  corresponds,  in  II ,  one,  and  but  one,  fundamental  point  together 
with  one,  and  but  one,  path  of  approach  to  it,  without  exception  ; 


ONE-TO-ONE   CORRESPONDENCE. 


25 


there  are,  therefore,  no  fundamental  points  on  F3  and  no  fundamental 
curves  in  the  plane  II. 

§  6.   Determination  of  all  straight  lines  on  jP3  . 
Let 

1  fa  +  02*2  +  £3tf3  +  £A  =  0, 

where 


(40) 


+  0, 


#1        ^2        03        ^^ 

be  any  straight  line,  I! ,  in  space  different  from  one  of  the  funda- 
mental lines  ;  the  parameters  A,  /x,  v  of  its  points  of  intersection 
with  the  surface  i^ ,  are  the  coordinates  of  the  points  of  intersection, 
different  from  the  fundamental  points,  of  the  two  cubics 

ai/i  +  aJ%  +  aJi  +  aJi  =  °> 
01^  +  02/2  +  03/3+04/4=0, 


(41) 


in  the  plane  II . 

If  the  line,  X',  is  to  lie  on  the  surface  Fit  these  two  cubics  must 
have  an  infinity  of  points  in  common ;  they  must  therefore  have  a 
common  factor  of  degree,  n  =  3 ,  2,  1 . 

1)  Let  <j>  be  a  common  factor  of  degree,  n  =  3,  in  \,  ft,  v . 
Set 

I  0x/  +  02/2  +  03/3  +  PJ*  =  P*4>,Pi  *  0; 

therefore 

(43)     foa,  -  *£)/,  +  foa2  -  p^,)/,  +  (p2«3  -  ^^)/3 

+   (^2«4~/3i04)/4=O, 

which  is  a  linear  relation,  in  which  on  account  of  (40)  not  all  the 
coefficients  vanish,  against  the  hypothesis. 

2)  Suppose  the  greatest  common  divisor  \jr  of  the  two  cubics  to  be 
of  degree,  n  =  2,  in  X,  fi,  v. 

Set 

Wi  +  02/2  +  03/3+  04/  =  +'  A> 
where  Lx ,  L2  are  linear  factors. 


26  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

Now  the  two  cubics  pass  through  the  six  fundamental  points, 
which  do  not  all  lie  on  a  conic.  Hence  the  conic,  i/r  =  0 ,  can  pass 
through  at  most  five  of  the  six  fundamental  points.  On  the  other 
hand,  not  less  than  five  of  the  fundamental  points  can  lie  on  ty  —  0  ; 
for,  if  there  were  four,  the  lines  LY ,  L2  would  pass  through  the  two 
remaining  points  and  would  therefore  coincide,  and  the  two  cubics 
would  have  a  common  factor  of  degree,  n  —  3 ;  if  there  were  less  than 
four,  three  or  more  fundamental  points  would  lie  on  a  straight  line. 

Hence  ty  must  pass  through  exactly  five  of  the  fundamental  points. 

Conversely,  if  ^r  =  0  be  any  conic  passing  through  five  of  the  six 
fundamental  points,  and  Lx ,  L2  are  two  distinct  straight  lines  inter- 
secting at  the  sixth  fundamental  point,  then  the  two  cubics,  sfr  •  Lx 
=  0 ,  sjr  ■  L2  =  0 ,  belong  to  the  linear  system  of  cubics  through  the 
points,  Pj ,  •  •  • ,  P6 ,  and  are  therefore  expressible  in  the  form  (  44 ) . 

To  the  points  of  the  conic  sfr  =  0  corresponds  then  a  straight  line 
on  the  surface  P3.  Now  six,  and  only  six,  different  conies  can  be 
described  to  pass  through  five,  at  a  time,  of  the  six  points,  PY ,  •  •  • ,  P6 ; 
hence  the  conic  i/r  =  0  can  occupy  six  different  positions  and  corre- 
spondingly, we  obtain  six  straight  lines  on  the  surface  P3 . 

3)  Let  the  greatest  common  factor  of  the  two  cubics  be  of  degree, 
n  =  1 ,  in  X,  //,  v. 

Set 

f  «i/i  +  <hf*  +  «s/s  +  aJi  =  L'Xi, 
(45)  \ 

where  %x ,  %2  are  quadratic  factors. 

Since  no  three  fundamental  points  lie  on  a  line,  at  least,  four  of 
them  are  external  to  L  —  0 ,  these  four  must  lie  on  %t ,  %  2  and  the 
remaining  two  must  lie  on  L  —  0 .  Conversely,  every  straight  line 
through  two  of  the  six  fundamental  points  gives  rise  to  a  straight 
line  on  the  surface.  We  thus  obtain  fifteen  straight  lines  on  the 
surface  P3,  which  together  with  the  six  fundamental  lines  and  the 
six  lines  found  under  b)  give  a  total  of  twenty-seven.  Therefore, 
there  are  twenty-seven,  and  only  twenty-seven,  straight  lines  on  the 
surface  P7, . 


CHAPTER   III. 

The  Transformation  of  an  Algebraic  Curve   from  the 
Plane  II  Upon  the  Cubic  Surface. 

§  7.   Generalities  about  cycles. 
Let 
(1)  K:         G(\,p,v)  =  0, 

be  an  algebraic  curve  in  the  plane  II ,  then,  according  to  the  theory  * 
of  the  singularities  of  algebraic  curves,  all  points  of  K  in  the 
vicinity  of  a  given  point  P0(\,  /*0,  v0)  of  K  are  expressible  by 
means  of  a  finite  number  of  convergent  power  series  of  an  auxiliary 
variable  t  (the  parameter)  as  follows  : 


(2) 


[  pv=vQ+ .  *,$  +  v2t2  H , 


called,  according  to  C.  Jordan  f  "  cycles."  Jordan  further  shows 
(Cours  D' Analyse  t.  I,  p.  563)  that  the  parameters  of  the  points  of 
intersection  of  a  straight  line 

(3)  A\  +  Bfi+Cv  =  0, 
with  the  cycle  ( C)  are  given  by  the  equation 

(4)  A\  +  Bfi0  +  Cv0  +  (A\  +  Bfi,  +  Cv^t  +  •  •  •  =  0. 

If  the  line  passes  through  the  point  P0 ,  the  "  center  of  the  cycle," 

we  have 

(5) A\  +  Bp,  +  Cvt-0, 

*Puiseux,  "  Recherches  sur  les  fonctions  algebriques,"  Liouville,  Journal  de 
Mathematiques,  t.  15,  p.  365  ;  Weieestrass,  "Theorie  der  Abelschen  Transcenden- 
ten,  Bd.  4,  p.  13 ;  Hamburger,  "  Ueber  die  Entwickelung  algebraischer  Func- 
tionen  in  Reihen,"  Zeitschrift  fur  Mathematik  und  Physik,  Bd.  16,  p.  461. 

fC.  Jordan,  "Cours  D' Analyse,"  t.  I,  p.  561. 

27 


28 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


and  if  \r,  ftr,  vr  is  the  first  column  of  coefficients  in  the  expansion, 
which  are  not  proportional  to  \,  //-0,  vQ)  the  expansion  (4)  begins 
with  the  term  (A\r  +  B\xr  -f  Ovr)tr.     Therefore, 


(«) 


(A\+Bpr+Cvr)f  +  ---  =  o, 


and  the  line  (3)  has,  atP0,  r  coinciding  points  in  common  with  the 

cycle  (C) ;  when  the  coefficients  A,  By  C  satisfy  besides  (5)  the 

equation 

(7)  AX  +jS/xr+  Cvr=0, 


the  line,  whose  equation  is 


W 


=  0, 


has,  at  P0 ,  more  than  r  coinciding  points  in  common  with  the  cycle. 
The  number  r  is  called  the  order  of  the  cycle  and  the  line  (8)  is  the 
tangent  to  the  cycle. 

If  the  point  JP0  is  an  ordinary  point  of  Ky  then  all  the  points  of 
K  in  the  vicinity  of  P0  are  expressible  by  one  single  cycle  of  order 
one  (a  linear  cycle).  If  P0  is  an  "  ordinary "  multiple  point  of 
order  o-,  all  the  points  of  Kin  the  vicinity  of  P0  are  expressible  by 
means  of  a-  linear  cycles  with  distinct  tangents. 

These  definitions  concerning  cycles  can  easily  be  extended  to 
curves  in  space  ;  let 


00 


Pxl  =  K0  +  K1t  +  K2t2+-- 

Pxs  =  Jfe  +  MJ  +  MJ+  • 

pxi  =  N0+Nlt  +  N2?+-. 


m 


be  a  cycle  in  space,  where  KQ ,  L0 ,  M0 ,  JV~0 ,  the  coordinates  of  PJ, , 
the  center  of  the  cycle,  are  not  all  equal  to  zero.  Then  the  pa- 
rameters of  the  points  of  intersection  of  an  arbitrary  plane 


(10) 


Ax1  +  Bx2  +  Cx3  +  Dxt  =  0, 


TRANSFORMATION  OF  AN  ALGEBRAIC  CURVE. 


29 


with  the  cycle  (  C  )  are  given  by  the  equation 

(11)  AK.+BL^  CM0+DN0+(AKl+BLl+  CMl+BNl)t+  . .  -  =0. 
If  the  plane  passes  through  the  center  P'0  of  the  cycle  (  C) ,  then 

(12)  AK„  +  BL0  +  CM,  +  BN0=0, 

and  if  Krf  Lr,  Mr,  Nr  is  the  first  column  of  coefficients  in  the  expan- 
sion (11),  which  are  not  proportional  to  KQy  L0,  MQ,  N0,  the  expan- 
sion begins  with  the  term  (AKr  +  BLr  +  CMr  +  DNr)tr .    Therefore, 


(13) 


(AKr+BLr  +  CMr  +  DNr)r  +  ...  =  0, 


and  the  plane  (10)  has,  at  P'0,  r  coinciding  points  in  common  with 

the  cycle  (C);  when  the  coefficients  A,  B>  C,  D  satisfy  besides  (12) 

the  equation 

(14)  AK  +  BLr  +  CMr  +  JDNr  =  0, 

where 


(15) 


K     L 


l0         ^'0 


*o, 


then  we  can  solve  equations  (12)  and  (14)  for  two  of  the  quantities, 
say  A  and  B,  in  terms  of  the  others,  C  and  D ;  hence,  by  substi- 
tuting the  values  thus  obtained  in  (10),  we  get  a  pencil  of  planes, 
whose  common  axis  is  given  by  the  equations 


(16) 


xl 

K 

K 

X2 

h 

4 

=  0, 

XS 

x. 

Mr 

K,     Kr 


K 


L 

r 

N 


=  0; 


and,  in  which  case,  the  line  (16)  has,  at  P'Q)  more  than  r  coinciding 
points  in  common  with  the  cycle  (C).  The  number  r  is  called  the 
order  of  the  cycle,  and  the  line  (16),  in  parameter  representation 


(17) 


is  the  tangent  to  the  cycle. 


px2=L0  +  Lru; 
px4  =  N0  +  Nru, 


30 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


§  8.   Transformation  of  the   non-singular  points   of  the 
algebraic  curve  k  to  the  cubic  surface. 

We  now  apply  the  results  of  the  preceding  sections  to  the  resolu- 
tion of  the  singular  points  of  an  algebraic  curve  into  double  points. 

Let 
(18)  K:         G(\ffi,v)  =  0, 

be  the  algebraic  curve  in  the  plane  LT ;  whose  singularities  we  wish 
to  resolve.  We  suppose  that  the  curve  K  is  irreducible  and  that  it 
has  r  ordinary  multiple  points,  Alf  •  •  • ,  Ar ,  with  only  linear  cycles 
and  with  distinct  tangents,  and  no  other  singular  points.*  We  may 
further  suppose  without  loss  of  generality  that  the  degree  of  the  curve 
K  is  greater  than  three. 

We  let  one  of  the  six  fundamental  points  coincide  with  the  multi- 
ple point,  Alf  to  be  resolved,  but  choose  all  other  fundamental  points 
external  to  K.  The  image  of  if  is  a  curve  K'  on  F3;  we  propose 
to  study  the  properties  of  this  curve  K\ 

Let  P'0  be  any  point  of  the  curve  K ';  according  to  §  5,  P'0  is  the 
image  of  one,  and  but  one,  point  P0  of  K.  The  point  P0  is  either 
an  ordinary  point,  or  one  of  the  multiple  points,  A2,  •  •  •,  Ar>  or  the 
multiple  point  Ax .  If  P0  is  an  ordinary  point,  then  all  points  of  K 
in  the  vicinity  of  P0  are  represented  by  one  linear  cycle  (  C) , 

/>\  =  \04- V  + V2+  •••> 

PI*  =  1*0  +  ^1  +  /*,*  +  .  ..,  (C) 

pv=Vt  +  Vlt  +  v2t2  +  -.., 
where  not  all  of  the  quantities  X0 ,  fiQ ,  vQ  equal  zero  and 


(19) 


(20) 


/*n 


+  0. 


(21) 


The  image  of  the  cycle  (C)  is  a  cycle  (C), 
[P'xl  =  K,  +  Klt+K2?  + 
p'x2  =  LQ  +  L1t  +  L2t2+. 
p'xz  =  M(i  +  Mlt  +  M2t2  + 
p'xi  =  N0  +  N1t+N2t2+. 


(C) 


*Noether,  Mathernatisehe  Annalen,  Bd.  9,  p.  166. 


TRANSFORMATION   OF   AN    ALGEBRAIC   CURVE. 


31 


a  cycle  of  the  curve  K' ;  where 

A  =fl  A  =  \/»  +  Kfl  +  *J%%  ■ 


(22) 


Since,  according  to  our  choice  of  the  fundamental  points,  JP0  is  in  the 
present  case  not  a  fundamental  point,  not  all  of  the  quantities,  KQ, 
A»  &.,  N0,  equal  zero. 

Suppose  iT0  4=  0.  We  propose  to  prove  that  the  cycle  (C)  is 
again  a  linear  cycle. 

For  if 


(23) 

we  should  have 


M0    iV0 


K.    Lo 


IsT,     Z,     J/,     JV, 


=  0, 


(24)  JT^p"^,      A -/A.      Xx>-P"X*>      N^p'N,, 

or  by  applying  Euler's  theorem, 

(25)  ( p\  -  Xx)fl  +  (p>0  -  j^)/*,  +  (p"„0  -  ^/J,  «  0 

(.7  =  1,2,3,4). 

The  rank  *  of  the  matrix  of  the  coefficients  in  these  equations  is 
r  =  3  ;  hence, 

(26)  p\-\l  =  0,       p"^-^  =  0,       p\-Vl=0, 
therefore, 


(9 


<) 


Xi      /*i      vi 


=  0, 


against  the  assumption  that  (C)  is  a  linear  cycle.     Therefore,  we 
must  have 


(28) 


K0    L0     M,    iV0|| 
K,     Lt     Mx     JV, 


+  0; 


*See§3. 


32 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


that  is  ( C )  is  a  linear  cycle  of  the  curve  K' ;  hence  the  image  JP'0  of 
an  ordinary  point  P0of  K  is  an  ordinary  point  of  K' . 

§  9.   Transformation  of  the  multiple  points  A2 ,  •  ■  • ,  Ar  to 

THE   CUBIC   SURFACE. 

If  P0  is  one  of  the  multiple  points,  A^ ,  •  •  •• ,  Ar ,  of  order  of  multi- 
plicity <r ,  then  all  points  of  K,  in  the  vicinity  of  JP0 ,  are  represented 
by  <t  linear  cycles  with  coinciding  centers  and  with  distinct  tangents. 

Let  (19)  and 

p\  =  \0  +  \;*  +  \;*2  +  ..., 
*w-v+X« +  «£*  +  '•■•, 

be  two  of  these  cycles,  then 


(29) 


(<y 


(30) 


\     P,      "c 


K  K 


+  0, 


and  since  the  tangents  to  the  two  cycles  are  distinct,  we  must  have, 
besides. 


(31) 


\     H     va 


/*i 


4=0; 


the  image  (C[ )  of  (Q)  is  given  by  the  equations 

A'asi  —  i0  +  i;<  +  i;<2  +  ---, 

p'xi=N!l  +  N'lt+N'2t>+..., 
where  F|  =  \;/JA  +  K/{„+  v[fl,  etc. 

Suppose  (C)  and  (C[)  have  coinciding  tangents ;  that  is 


(32) 


(<?n 


(33) 


iT0     Z,     Jf,     N, 

Kt     Lx     Jf,     iV, 

ir;   z;   i/;   iv; 


=  o 


or  THE 

UNIVERSITY 

OF 
TRANSFORMATION   OF   AN   ALGEBRAIC  CURVE.  33 

Consider  one  of  the  four  equations  implied  in  (33) ;  as 


(34) 


K„     X„    M„ 


l: 


if, 

m: 


=  o 


On  account  of  (22)  and  using  Euler's  theorem,  we  obtain  from  (34) 


(35) 

But  by  (31) 

therefore 
(36) 


J  \\ 
J  it*. 

fl 


fl 
fl 


J  2A 
./  2M 

f° 
J  2v 


f  2\ 
J 2H- 
J  2v 

4=0, 


f  3X 

./  3M 


=  0 


./3A 

J  3M- 
y  3v 


=  0, 


and,  in  the  same  way,  it  follows  that  all  the  determinants  of  order 
three  of  the  matrix, 

(37)  11  /i  r„  /mi, 

are  equal  to  zero ;  but,  according  to  §  3 ,  this  is  impossible,  since 
also  in  the  present  case  P0  is  a  non-fundamental  point.  All  the 
points  of  K'  in  the  vicinity  of  P'Q  are  therefore  represented  by  a 
linear  cycles  with  coinciding  centers  and  with  distinct  tangents  ;  that 
is,  the  image  A',  of  one  of  the  multiple  points  A.  (i  —  2,  3,  •  •  •,  r) 
of  multiplicity  <r.  is  an  ordinary  multiple  point  of  ordei*  <r.  of  K'. 

§10.  Transformation  of  the  multiple  point  Ax  to  the 
cubic  surface. 

If  the  point,  P0 ,  is  the  multiple  point  Ax ,  of  any  order  of  multi- 
plicity <rx ,  then  all  points  of  K  in  the  vicinity  of  P0  are  represented 
by  <r1  linear  cycles  with  coinciding  centers  and  distinct  tangents. 


34 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


According  to  our  choice  of  the  fundamental  points,  the  point  P0 
or  Ax  coincides  in  this  case  with  one  of  the  fundamental  points. 
Hence  if  (19)  represents  one  of  the  cr1  cycles,  (C),  with  center  P0 
and  (21)  its  image  ( (7)  on  F3 ,  then 


(38)  Jf0=0,         Z0  =  0,         Jf0  =  0,        iV0  =  0, 

and  after  division  by  t  the  cycle  (  C)  takes  the  form 


(39) 


(<?') 


Pxt=N1  +  Nit+N3t?  + 


We  propose  to  show  that  this  cycle  is  again  a  linear  cycle,  there- 
fore that 

II  Kl    Ll    Ml    Nt 


(40) 


K%    i2    3ft    N2 


+  0. 


(41) 


In  order  to  prove  it,  we  suppose 

-K,     Lx    Mx    N, 


K,    h    M2    Nt 


=  0; 


the  quantities  KY ,  Lx ,  Ml ,  2V,  are  not  all  equal  to  zero,  as  has  been 
proved  in  §  4,  b);  suppose  KY  4=  0,  then  (41)  is  equivalent  to  the 
three  equations, 

(42)  K,L2-K2LX  =  §,  KxM2-K2M,  =  0,  K^- K2Nl  =  0. 

For  the  further  discussion  of  these  quantities,  we  suppose  the  fun- 
damental point  with  which  P0  coincides  to  be  P3 ,  and  we  make  use 
of  the  special  system  of  coordinates  and  the  notation  introduced  in 
§  5 ,  d)  ;  hence,  we  have 


TRANSFORMATION   OF   AN   ALGEBRAIC    CURVE. 


35 


(43) 


iT2  =  Px{bx\^  +  6a^  +  2^*,  +  ^2)  -  ftft.A^, 

^2  =  %(^\ft  +  ciM  +  2os  Vi  +  C3X2)  -  7a%\^i  J 
and  these  values  substituted  in  (42)  give 

ft7i(«iM!  -  <*2\)  -  «2A63C3X1  =  0> 

XiM1[/3172(>«i«2  -  A*i«i)  +  W08y»*i  -  ^i7s^)] 

+  ^l7263C3(X2^1  -  \^2)  =  °J 

Since  we  supposed  i£i  4s  0 ,  therefore,  /ax  +  0 . 
In  IT,  if  \  =  0,  then  p,  =  0,  against  the  assumption 

0      0       1 

4=  0 ;  hence,  \  4=  0 


(44) 


I; 

II; 
III. 


(45) 


Mi 


By  equating  the  values  of  \/*2  —  X^  obtained  from  I  and  III,  we 

have 

(46)  V2/3272  -  M1(a,63c3A  +  /S,?,*,)  =  0; 


and  by  II ,  therefore 

\(a2Ab3c3  +  ^7,0,)  -  fi^ff^  =  0; 

a2&2%  «l^C3A  +  ^2  72ai 

a263C3A+a2^l7i  O^Ti 

^^l^l  +  Oj^^^a  +  «1«263C3A  -  °> 


(47) 
hence 

(48) 


0, 


therefore 

(49) 

and  finally, 


(50) 


ax  6X     ax  Cj     6j  cx 
«262      a2c2      M2 

«363       a3C3       63C3 


o, 


36  SINGULARITIES   OF   ALGEBRAIC  CURVES. 

the  necessary  and  sufficient  condition  that  all  six  fundamental  points 
shall  lie  on  a  conic,  against  the  hypothesis.  Hence,  the  cycle  (  C )  of 
the  curve  K'  is  a  linear  cycle. 

According  to  §  4,  the  centers  of  the  <rx  linear  cycles  of  the  curve 
K'  are  all  distinct,  distributed  along  the  fundamental  line,  each 
center  being  determined  by  the  tangent  line  to  the  corresponding 
cycle  of  the  curve  K;  and  according  to  §  5,  each  center  of  the  <rl 
linear  cycles  of  K'  does  not  have  the  center  of  any  other  linear 
cycle  to  coincide  with  it.  Hence,  each  point  of  the  <rx  points  of  K'  is 
an  ordinary  point.  Therefore  the  curve  K  is  transformed  into  a 
new  curve  K'  on  the  surface  Fz ,  given  by  the  equations 


(51) 


px±  —f(\,  /jl}  v),  combined  with  the  equation 


which  has  the  multiple  point  Ax  of  K  resolved  into  ordinary  points  and 
all  the  remaining  multiple  points  A2,  . . . ,  Ar  of  K  transformed  into 
new  multiple  points  A2}  •  •  •,  A'r  unchanged  in  orders  of  multiplicity f 
with  linear  cycles  and  with  distinct  tangents,  and  without  the  introduc- 
tion of  any  new  multiple  points  or  other  singularities. 


CHAPTER  IV. 

Transformation  from  the  Surface  Fz  to  a  Plane  IT. 

§  11.   One-to-two  correspondence  between  the  surface  Fs 
and  the  plane  it. 

We  now  select  a  point  0  on  F3 ,  and  a  plane  II '  not  passing  through 
0 ,  and  project  K'  from  0  to  II ' ,  thus  obtaining  a  plane  algebraic 
curve  K".  We  are  going  to  prove,  that  the  point  0  can  be  so  chosen, 
that  the  correspondence  thus  established  between  the  curves,  K'  and 
K",  is  a  one-to-one  correspondence  of  points  and  that  the  curve  K" 
has  r  —  1  ordinary  multiple  points,  A'2i  •  •  •,  A"ri  of  the  same  multi- 
plicities as  the  points  A2 ,  •  •  • ,  Ar  of  ^T,  and  besides  no  other  mul- 
tiple points,  except  ordinary  double  points. 

Formulae  for  the  correspondence  between  F3  and  II ' : 
We  introduce  a  new  tetrahedron  of  reference,  one  of  whose  vertices, 
f  =  0 ,  17  =  0,  f  =  0 ,  t  =  1,  coincides  with  the  point  0,  while  the 
opposite  face  ABC,  t  =  0,  coincides  with  the  plane  II '.  If  £0,  rjQ, 
f0 ,  t0  are  the  coordinates  of  any  point,  P'Q ,  on  F3  different  from  0 , 
the  projecting  ray,  OP  'Q ,  is  given  in  parameter  representation  by  the 
equations, 

(1)  />£='i£0>         PV  =  tlVoy         p^=t^Qf         /jt  =  ^t0-}-^. 
The  coordinates  of  the  projection,  P'0' ,  are 

(2)  />£  =  £o>         PV  =  V0,         P?-  $»         t  =  0. 

The  quantities  f 0 ,  rjQ ,  f0 ,  which  are  not  all  three  zero,  are  at  the 
same  time  the  trilinear  coordinates  *  of  the  point,  jP'0',  with  respect 
to  the  triangle  of  reference  ABC  in  II'.  Only  when  the  point,  P'q, 
coincides  with  0  does  its  projection  become  indeterminate.  In  order 
to  avoid  this  exceptional  case  in  the  projection  of  K\  we  impose, 
upon  the  point  0,  condition  I:  that  the  point  0  shall  not  lie  on  the 
curve  K' . 


*  Clebsch-Lindemann  "  Vorlesungen  uber  Geometric,"  Bd.  II,  p.  99. 

37 


38  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

Vice  versa  :  If  £0 ,  nQ  ,  f0  are  the  trilinear  coordinates  of  any  point 
PJJ  in  II'  with  respect  to  the  triangle  of  reference  ABC,  then  the 
tetrahedral  coordinates  of  the  point  PJ,  are  ?o  >  vo  i  ?o  ?  ^  an^  ^he  Pro" 
jecting  ray  OP'Q  is  given  by  the  equations, 

(3)        />?  =  £<A>       n**%h>       &*~ZAt       pT  =  t2. 

If  we  substitute  these  values  in  the  equation  of  the  cubic  surface,  we 
will  get  a  cubic  equation  in  tx  :t2.  If  this  cubic  equation  does  not 
degenerate  into  an  identity  its  three  roots  will  give  the  parameters 
of  the  three  points  of  intersection  of  the  ray  OP"  with  Fz .  One 
of  these  points  is  0 ,  the  two  other  points  we  denote  by  P\  and 
P'2 .  If  the  cubic  equation  degenerates  into  an  identity,  it  is  satis- 
fied for  all  values  of  \  :t2,  and  therefore  the  ray  lies  wholly  on  Fs. 
In  order  to  avoid  the  exceptional  case,  we  impose,  upon  the  point 
0,  condition  II:  that  the  point  0  sliall  not  lie  on  any  one  of  the 
twenty -seven*  right  lines  on  Fz.  We  thus  obtain  a  one-to-two 
correspondence  of  points  between  F3  and  II' ;  the  only  exception  is 
the  point  O  on  P3 ,  to  which  corresponds  in  the  plane  II'  the  line  of 
intersection  of  the  tangent  plane  f  to  FB  at  0  with  the  plane  II'. 
The  locus  made  up  of  the  twenty-seven  right  lines  on  F3 ,  we  denote 
byA,. 

§  12.  Kelated  points. 

If  XJ,  fi°v  v\  and  \\,  /*§,  v\  are  the  parameters  of  the  two  points 
P[  and  P2  on  F3 ,  and  if  P[  does  not  coincide  with  0  and  does  not 
lie  on  one  of  the  six  fundamental  lines,  then  \° :  \t*\  '•  v\  are  rationally 
expressible  in  terms  of  \\ ,  p\ ,  v\ . 

The  coordinates  of  P  [ ,  in  the  new  system,  are 

where  f\ ,  f2 ,  /J ,  f\  are  the  same  linear  combinations  of  flf  f2,fif 

*See§6. 

f  There  exists  a  definite  tangent  plane  to  F3  at  0,  since  0  is  not  a  singular  point 
of  Ft  (see  §  5). 


TRANSFORMATION  FROM  SURFACE  TO  PLANE. 


39 


f4  as  f ,  y,  f,  t  are  of  a?t,  #2,  #3,  z4 ;  hence  the  equations  for  the 
projecting  ray  OP[  become 


(5) 


If  X ,  /a  ,  v  are  the  parameters  of  one  of  the  points  P '  of  intersection 
of  OP[  with  JFl ,  then  the  coordinates  of  P'  are 


(6) 


Since  P'  lies  at  the  same  time  on  OP\  it  must  be  possible  to  so 
determine  tx :  t2 :  £3  that 


(7) 


(8) 


=  0, 


«,/;(M»^»»j)^<i/i(*»M,*), 

To  obtain  the  points  of  intersection,  (7)  must  be  solved  with  respect 
to  \:  fi:v  and  ^ :  ^  :  tt , 
From  (7)  we  get 

A(K,rt,<)  fAK,ti,>V  AiH'^*) 

/;(x,m,j')    /;(*,>>*)    /i(^c') 

not  all  of  the  quantities, 

aw,  m?,  >&/;<*;>  *•:,  <),/;(*•:>  *?>  o-, 

are  equal  to  zero,  since  P\  is  different  from  0  and  does  not  lie  on 
one  of  the  six  fundamental  lines. 

Set/i(Xj,  fx°iy  vl)  4=  0;  then  (8)  is  equivalent  to  the  two  equations 

These  two  cubics  can  not  contain  any  common  factor,  because  OP'l  can 


40  SINGULAKITIES   OF   ALGEBRAIC  CURVES. 

not  lie  wholly  on  Fs,  since  0  does  not  lie  on  one  of  the  twenty-seven 
straight  lines  of  F3 ,  they  intersect  therefore  in  nine  points ;  eight  of 
these  points  are  the  points,  Px,  •  •  •,  P6,  (XJ,  /i°lf  i>J),  and  a  point 
(  X° ,  fxQ ,  v° )  ,  which  is  the  image  of  0  in  IT  .  The  ninth  base  point 
P°2(\°2,  fi°2 ,  v\ )  is  uniquely  determined,  and  therefore  X° :  fi°2 :  v\  are 
rational  functions  of  Xj :  p\ :  v\ ;  say,  * 

(10) rf>x2-*(x«, #,*?),  K=f(\j,K,"?),  rt~#(\',rf.*0> 

where  <£ ,  ^ ,  %  are  homogeneous  integral  functions  of  Xf ,  ft  J ,  ^J . 

§  13.   One-to-one  correspondence  between  the  curves  K 

and  K" . 

Although  the  projection  from  the  point  0  establishes  a  one-to- 
two  correspondence  between  the  surface  FB  and  the  plane  II',  never- 
theless the  correspondence  between  the  two  curves  K'  and  K"  de- 
fined by  the  same  projection  is  a  one-to-one  correspondence  provided 
the  curve  K  in  the  plane  II  is  irredueible,  as  we  have  supposed. 

In  order  to  prove  it,  we  consider  any  point  P\  of  K'  and  ask 
under  what  conditions  will  the  related  point  P'2  to  P[  also  lie  on  K'l 

The  curve  K' ,  in  the  new  system  of  coordinates,  is  given  by  the 
equations 

tpr=/;(X,  fi,v),   G(X,pL,r)~Q. 
If  P\  and  P'2  both  lie  on  K' ,  then  we  have 
(12)  G(\%tf,v<l)  =  0,     G(\l,^,vl)  =  0; 

the  latter  equation  according  to  §  12,  may  be  written 

(13)  gwx> ti> 0>  +(K A  *)>  x(K rt> "M  -  Gi(K, K, *J)  =  o. 

*By  an  easy  limiting  process  it  can  be  shown  that  the  formulae  (10)  still  hold  for 
the  excluded  points,  with  the  understanding  that  the  ratios  <?> :  V  :  X  are  replaced  by 
their  limits  as  the  point  ( \°,  ^° ,  vt°)  approaches,  along  a  given  path,  one  of  the  ex- 
cluded points. 


TRANSFORMATION   FROM   SURFACE   TO   PLANE.  41 

Hence  the  coordinates  must  satisfy  simultaneously  the  two  equations 

(14)  &(%,$,  4)-Q,     ffiWiAO.-'O, 

if  P'2  lies  on  K' . 

The  equation,  ^(Xj,  /*J,  vj)=  0,  can  not  reduce  to  an  identity 
holding  for  all  values  of  \J ,  fi%  v\;  for,  let  $'  be  any  point  on  Fs , 
different  from  0  and  not  on  K' ,  if  we  choose  for  P\ ,  the  third 
point  of  intersection  of  OQ'  with  F3,  then  the  point  P'2  coincides 
with  (J)' and  therefore  does  not  lie  on  K' ,  and  hence  Gx  (  \\ ,  /ij ,  v\ )  =f=  0. 

We  now  make  use  of  the  hypothesis  that  K  is  irreducible  *  ;  then 
the  two  equations 

(15)  G(\,H.,v)  =  0,      G1(\)H.,v)  =  0, 

have  either  no  common  factor,  or  the  factor  G(\,  p,  v).  In  case 
first,  G  and  GY  can  intersect  in  only  a  finite  number  of  points  ;  hence, 
there  can  be  only  a  finite  number  of  points  P\  on  K'  for  which  P'2 
falls  on  iT' .  In  case  second,  for  all  points  P[  on  K'  the  related 
points  P2  lie  also  on  K' . 

Hence,  if  we  can  find  one  particular  position  of  P\  for  which  P'% 
does  not  lie  on  K' ,  then  we  have  case  first. 

Now,  the  tangent  plane  to  Fs  at  0  meets  K'  in  at  least  one  point, 
say  P[ ,  and  the  ray  OP\  meets  Fz  in  the  three  points,  P[ ,  0 ,  0 ; 
therefore  the  point  P'2  must  be  the  point  0  and  hence  does  not  fall 
on  the  curve  K' ,  since  0  f  is  not  on  .ST .  There  exists  therefore 
only  a  finite  number  m  of  pairs  of  related  points,  P\ ,  F2 ,  which  lie 
both  on  K' ;  the  projections  of  two  such  related  points  coincide  in  a 
point  of  the  curve  K"  and  give  rise  to  a  new  multiple  point  of  K" . 
These  m  new  multiple  points  will  be  denoted  by  D'[y  •  •  •,  D"m  .  To 
every  other  point  of  K"  corresponds  one,  and  but  one,  point  of  K', 
and  in  this  sense,  the  projection  from  the  point  0  establishes  indeed  a 
one-to-one  correspondence  between  the  two  curves  K'  and  K" . 

*See§8. 
f  See  §  11. 


42 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


(16) 


§  14.  The  orders  of  the  cycles  of  the  curve  K" . 
Let 

^  =  £o  +  fi*  +  f/  +  ---, 
PV=V0  +  Vit  +  Va*  +  •••, 

^T  =  To  +  Ti^  +  T2^2+  •••> 


(C0) 


be  a  cycle  of  the  curve  K'  expressed  in  the  new  system  of  coordi- 
nates. Since,  according  to  §§8,  9,  10,  all  cycles  of  K'  are  linear, 
we  have 


(17) 


0       ^0       '0       To 


£i      Vi      Si      Ti 


#0, 


where  not  all  three  of  the  quantities,  f 0 ,  t;0  ,  JJ, ,  are  equal  to  zero, 
since  the  center  of  (OJ)  is  different  from  the  point  0  (0,  0,  0,  1), 
which  is  not  on  K' . 

The  cycle  (C0)  is  projected  into  a  new  cycle  ( CJ)  in  II',  given 
by  the  equations 

'/>£  =  ?0  +  £i<  +  !/  +  ---, 
(18)  J  pv  =  v„  +  vlt  +  v^+---, 

.P?=  ?„  +  £,« +  ?/  +  •••. 

We  ask  under  what  conditions  will  (Cq)  be  of  higher  order  than 
the  first? 

This  will  be  the  case  if,  and  only  if, 


(19) 


%     V0      fo 

fi    ^i    Si 


==o. 


But  if  (19)  is  satisfied,  we  can  determine  two  quantities  kl9  k2, 
so  that 

(20)      0  =  ^  +  ^,     0  =  ^  +  ^,,     0  =  5^  +  ^, 

and  therefore  the  point  O  lies  on  the  tangent  to  the  cycle  (Cq), 


TRANSFORMATION   FROM   SURFACE   TO   PLANE. 


43 


which  may  be  written 


(21) 


pi  =  f0*i  +  Zil2>    Pv  -  %h  4-  V2> 


If,  therefore,  we  construct,  in  every  point  P'  of  K' }  the  tangent 
to  K' ,  or,  in  case  P  should  be  a  multiple  point  of  JS7,  the  tangents 
to  the  linear  cycles  with  coinciding  centers  at  P'  and  denote  by  Q' , 
in  each  case,  the  third  point  of  intersection  f  of  the  tangent  with  F3, 
then  as  the  point  P  describes  the  curve  K'  the  point  §'  will  de- 
scribe a  curve  on  Fz ,  which  we  denote  by  A2 .  If  now  we  impose 
condition  III  upon  the  center  of  projection  O,  that  the  point  0  shall 
be  external  to  this  locus  A2 ,  then  linear  cycles  of  K'  are  projected 
into  linear  cycles  of  K" ,  and  since  all  the  cycles  of  K'  are  linear, 
also  all  the  cycles  of  K"  will  be  linear. 

§15.   The  projection  of  the  multiple  points  A'tf  •••,   A'r 

of  K'. 

Let  (16)  and 

,*"■%+■««  +  <<■  + 

be  two  of  the  linear  cycles  with  the  multiple  point  A\  of  K'  for 
center,  then 

£o      %      £o       To 

?1       ^l       4,       T, 

and  according  to  §  9  (C^)  and  (C[)  have  distinct  tangents,  that  is 
f.      %      ?o      To 

fi     ^i     ?!     r, 


(CI) 


(23) 


+  0; 


(24) 


?1  *?!  ?1  Tl 


H=0 


*See§7. 

f  It  might  happen  that  the  tangent  lies  wholly  on  the  surface  F3 ,  in  which  case 
it  would  coincide  with  one  of  the  twenty-seven  straight  lines  of  F3 .  But  since  we 
have  already  imposed  upon  0  the  condition  not  to  lie  on  one  of  these  twenty-seven 
straight  lines,  no  special  provision  is  necessary  for  this  contingency. 


44  SINGULARITIES   OF   ALGEBRAIC   CURVES. 

The  new  cycle  {C'()  of  K"  is  given  by  the  equations 

(25)  <   PV=V0  +  v'lt  +  v'2t2+--, 

The  two  cycles  (GJ )  and  (0'/)  will  have  coinciding  tangents  if,  and 
only  if, 


(26) 


f. 

% 

?o 

1, 

V, 

r, 

?; 

n'i 

c 

=  0. 


This  equation  has  a  simple  geometrical  meaning :  the  equation  of  the 
plane  passing  through  the  tangents  to  (OJ)  and  (C[)  is,  according 
to  §7, 

Z         V         t         T 


(27) 


0 


?o     %      ?o      To 
£i     ^i      ?i      Ti 

?i    ^i    Si    Ti 

now  if  the  point  0  lies  in  the  plane  (27),  we  have 


(28) 
that  is 


0 

0      0 

i 

fa 

%      ?o 

To 

f, 

^i      ?i 

Tl 

£ 

>?;  s 

=  0, 


?o      %      ?o 

fl   *1    Si 

fi   *i   Si 

This  plane  (27)  is  the  tangent  plane  to  Fs  at  P'0;  for  the  tangents 
to  the  cycles  (C'0)  and  (C[)  are,  at  the  same  time,  tangents  to  the  sur- 
face F3 ;  and  since  F3  has  a  definite  tangent  plane  at  A\  (see  §  5), 


TRANSFORMATION  FROM  SURFACE  TO  PLANE.        45 

this  tangent  plane  must  be  identical  with  the  plane  through  the  two 
distinct  tangents  to  the  cycles  (C'Q)  and  (C{). 

The  plane  (27)  cuts  the  surface  F3  in  a  curve  H.  Hence,  if  we 
construct  tangent  planes  to  F3  at  each  multiple  point  A2 ,  •*  •  \  A'r  of 
K' ,  we  shall  obtain  r  —  1  curves  H2y  •••,  Hr,  which  constitute 
together  a  locus  which  we  denote  by  A3 ;  and  if  we  impose,  upon 
the  center  of  projection  0,  condition  IV,  that  O  shall  be  external  to 
this  locus  A3,  then  for  each  one  of  the  multiple  points  A2,  •  •  •,  A'r 
of  K'  the  a.  linear  cycles  of  A\  will  be  projected  into  a.  linear  cycles 
with  the  same  center  A\  and  imth  distinct  tangents. 

From  the  conditions  I— IV  imposed  so  far  upon  the  point  0,  it 
follows  therefore  that  if  P"  be  any  point  of  K"  different  from  the 
m  points  DJ',  .  •  •,  D£f  then  either  P"  is  the  projection  of  an  ordinary 
point  of  K' y  in  which  case  P"  is  again  an  ordinary  point  of  K"\  or 
P"  is  the  projection  of  one  of  the  multiple  points  A\  (i  -«■  2,  •  •  • ,  r), 
with  <t.  linear  cycles  with  distinct  tangents,  in  which  case  P  "  is  an  ordi- 
nary multiple  point  of  K"  with  the  same  number  o\  of  linear  cycles 
and  with  distinct  tangents. 

It  remains  therefore  only  to  study  the  points  D'[ ,  •  •  • ,  D"m  of  K" . 

§  16.   Condition  to  prevent  the  points  D"  from  being  of 

HIGHER   MULTIPLICITY  THAN  THE   SECOND   ORDER. 

In  each  point  D"  coincide  the  projections  of  two  related  points 
P[ ,  P'2  of  K' .  If  the  two  points  P\ ,  P2  are  ordinary  points  of 
K' ,  the  point  D"  will  be  a  double  point  of  K' ' ;  if  one  of  them  is 
one  of  the  multiple  points  A'. ,  the  point  D"  will  be  a  multiple  point 
of  K"  of  higher  multiplicity. 

The  latter  case  will  present  itself  if,  and  only  if,  the  third  point 
of  intersection  of  the  line  OA \  with  the  surface  Fs  lies  on  the  curve 
K\  Hence  the  projection  of  A',  can  not  coincide  with  the  image 
of  any  other  point  P0  of  K' ' ,  if  the  point  0  does  not  lie  on  the  curve 
T.  described  by  the  third  *  point  of  intersection  Q  of  the  line  A'.P0 

*  For  particular  positions  of  the  point' P/  the  line  A/P/  might  lie  wholly  on  the 
surface  J^8  ;  the  center  0  will  then  certainly  not  lie  on  the  line  A/Pq',  since  we 
have  supposed  that  0  shall  not  lie  on  one  of  the  straight  lines  of  the  surface  Fs . 
See  a  11. 


46 


SINGULARITIES    OF    ALGEBRAIC   CURVES. 


with  F3  as  the  point  P'0  describes  K' .  If  we  denote  by  A4  the  locus 
made  up  of  the  r  —  1  curves  r2,r3,  •  •  • ,  Tr  and  impose  condition  V 
upon  the  point  0,  that  0  shall  be  external  to  this  locus  A4,  then  the 
image  of  none  of  the  multiple  points  A'2 ,  •  •  •,  A'r  will  coincide  with 
the  image  of  any  other  point,  simple  or  multiple,  of  the  curve  K '.. 
The  points  D'[,  -  • .,  If'm  will  therefore  be  double  paints  of  K" ;  and 
on  the  other  hand,  if  A\  is  the  center  of  <r.  linear  cycles  of  the  curve 
K' ,  then  the  image  A\  of  A\ ,  in  the  plane  II',  will  likewise  be  the 
center  of  a.  (and  not  more)  linear  cycles. 


§  17.   Condition  to  prevent  coinciding  tangents  at  the 

DOUBLE   POINTS   D'[,  •  •  • ,  IT. 

Let  Jf  be  any  one  of  the  double  points  D'[  •  •  •,  Df'm  and 
•^i  (  £ i  *  ^i>  ?i  9  Ti )  anc*  &*{  f 2  9  Vz ,  £j  >  T2 ) tne  two  related  points  of  Fz , 
both  lying  on  K' ',  whose  projections  P^(|, ,  Vi y  ?i )  ana<  P'i (  &>  %>  (*) 
in  the  plane  II'  coincide  in  the  point  D",  so  that 


(29) 


ii  Vi  ^ 
&  %  f, 


=  0, 


at  the  same  time  not  all  three  of  the  quantities  fu  ^,  £,  nor  f2,  ?72, 
£2  can  be  equal  to  zero,  since  on  account  of  condition  I  neither  P[ 
nor  P  2  coincides  with  0 .  Further  the  points  P  [  and  P  J  are  distinct, 
according  to  §  13,  hence 


(30) 


£     'h     ?i     Ti 


+  0. 


»2        ^2         '2  2 

According  to  §  16 ,  P{  and  P2  are  both  ordinary  points  of  Kf ;  let 


(31) 


(CD 


TRANSFORMATION    FROM   SURFACE   TO   PLANE. 


47 


and 


(32) 


>?  =  &  +  £<  + 
pv  =  v2  +  v'2t  + 

»  PT=T2+T'it  + 


(<?3 


be  the  two  linear  cycles  which  represent  the  curve  K'  in  the  vicinity 
of  P{  and  P'2  respectively. 

The  projections,  in  the  plane  II ',  of  these  cycles  are 


(33) 
and 
(34) 


**-*,  +  £<  + 

pv  =  v1  +  v',t  + 

ft-  £  +  ?;<  + 

PV  =  V2  +  v'2t  + 

l />?=?,  +  ?;<  + 


(C?) 


(c;) 


respectively  ;  their  coinciding  center  is  the  point  f  t :  rjl :  fj  =  f  2 :  rj2 :  £2 . 
The  tangents  to  the  two  cycles  (C")  and  (C'2')  will  coincide  if,  and 
only  if, 

£i    **    £i 

(35)  g  v[  r;  =0. 

?2        ^2         G 

This  equation  has  a  simple  geometrical  meaning :  the  equation  of  the 
plane  through  the  tangent  to  the  cycle  (C[)  and  the  point  P'2  is 


(36) 


f     V 

t      T 

*i  % 

t,    r, 

%  v[ 

G  *; 

fc  % 

?2      Tl 

=  0, 


the  condition  that  this  plane  (which  on  account  of  (29)  always  passes 


48  SINGULARITIES    OF   ALGEBRAIC   CURVES. 

through  0)  contains  the  tangent  to  the  cycle  ((72)  is 

£«  v2  Z  T2 

fl  %  ^  Tj 
fl  ^1  ?!  Tj 
fj      ^2       ?2       T2 


(37) 


since  the  point  g'2 ,  $j ,  Ja ,  T2  is  a  point  of  the  tangent  to  ( C2 ) . 
But  according  to  (29),  a  quantity  />4=  0  can  be  determined  so  that 


(38) 


f2  =  P?i> 


P? 


a 


?2  =  P?1 


Hence  if  we  subtract,  in  the  determinant  (37),  from  the  last  row  the 
second  multiplied  by  p  and  notice  that  t2  4=  prx  on  account  of  (30), 
we  obtain  (35). 

The  tangents  to  the  two  cycles  (C")9  (C2')  with  the  center  D"  will 
therefore  coincide  if,  and  only  if,  the  tangents  to  the  two  cycles  (C[) 
and  (C  2)  lie  in  one  plane  passing  through  0. 

We  must  now  impose  upon  the  point  0,  a  condition  which  will 
prevent  the  point  O  from  lying  in  a  plane  containing  two  tangents 
to  K' '.  For  this  purpose,  let  P[  be  any  ordinary  point  of  K' ,  and 
let 

(39)  &+;fo-o 

represent  the  pencil  of  planes  through  the  tangent  to  K'  at  P[ ;  f ,  77, 
being  homogeneous  linear  functions  of  f ,  77,  £,  t  . 

Let  P  2  be  another  point  of  K\  the  image  of  an  ordinary  point  P2 
of  K\  let 

(40)  J    /i  =  /i0  +  /i^+..-, 
I  */  =  *>0  +  V  +  •*•> 

be  the  linear  cycle  which  represents  the  curve  iT  in  the  vicinity  of 
P2 ;  then  the  series  (40)  satisfy  identically  the  equation  of  the  curve  K, 


(41) 


G(\,^v)m.O, 


TRANSFORMATION   FROM   SURFACE  TO   PLANE. 


49 


hence 

(42)  G(\,  AS,  vt)  =  0, 

(43)  \GX(\,  im0,  pt)  +  Pk$,(\j  a*0,  "„)  +  viGA\>  *•*>  vo)  =  °- 

The  tangeDt  to  K"  at  P'%  is,  according  to  §  8,  given  by  the  equa- 
tions 

\i*-fl  +  4t,      in-fl  +  At, 

where 


(45)     \Jj=Mx*>*»vo)  (i  =  l,2,3,4), 

Hence  if  the  tangent  to  K'  at  Pg  is  to  lie  in  the  plane  (39),  it  is 
necessary  and  sufficient  that 


(46) 


where  f\,  f\,  («?,  «°)  are  the  same  linear  functions  of  /J,  • 
(«},  •  •  •,  sj)  as  ?, »?  are  of  f ,  if,  K,  t.     Hence  we  must  have 


>  ft) 


(47)  7^-^  =  0, 

or,  if  we  make  use  of  Euler's  theorem, 

(/Vi-*n)(AA-AA)+(\*-*\KAA-AA) 
+(Vi  -  V.XAA  -A  A)  =  o- 

But  the  equation  (42)  may  be  written 

(49)    \GX(\,  fi0>  *0)  +  /*o#m(xo>  ^o>  *o)  +  "o^K*  A*o>  *o)*°0> 
therefore 


(48) 


(50) 


50 


SINGULARITIES   OF   ALGEBRAIC   CURVES. 


Hence  we  obtain  the  result :  If  the  tangent  to  K'  at  P'2  lies  in  one 
of  the  planes  of  the  pencil  (39)  then  the  parameters  \0,  fiof  v0  must 
satisfy  the  two  equations 


=  0, 


J  IK        JZk         «A 

(51) 

H(\,  /*,  v)  m 

fly.      fly.       Gy, 

and 

f\v        J2V         «V 

(52) 

G(\, 

M,^)  =  0. 

Vice  versa :  If  X0 ,  /*0 ,  v0  satisfy  these  equations  there  exists  a 
plane  of  the  pencil  (39),  which  contains  the  tangent  to  K'  at  fr 

Since  (52)  is  irreducible  the  two  curves  (51)  and  (52)  have  either 
a  finite  number  of  points  of  intersection,  or  else  H  is  divisible  by 
G,  in  which  case  every  tangent  to  K'  is  contained  in  a  plane  of  the 
pencil  (39). 

In  the  latter  case,  the  curve  K'  would  be  a  plane  curve.  To 
prove  it,  we  use  for  greater  convenience  non-homogeneous  rectangular 
coordinates  x ,  y ,  z,  the  s-axis  coinciding  with  the  tangent  to  Kf  at 
PJ,  and  origin  with  the  point  P[.  If  the  curve  K'  is  given  in 
parameter  representation  by 

(53)  »-*(*),      y  =  *(0>      »,***(*)» 

and  if  the  tangent  at  the  point  t 

X-x      Y-y      Z-z 

(54)  j—  = r1  m  —7-  , 

v     '  x  y  z      ' 

is  to  pass  through  the  2-axis,  we  must  have 

(55)  ^'-^'=0, 

and  if  this  is  true  for  every  tangent,  then  we  obtain,  by  integrating 
(55), 

(56)  y  =  cx, 

where  c  is  a  constant.     That  is,  the  curve  K'  lies  in  a  plane  passing 
through  the  s-axis. 


TRANSFORMATION  FROM  SURFACE  TO  PLANE.        51 

If  the  curve  K'  is  a  plane  curve,  the  curve  K  is  a  curve  of  the 
system  of  cubics  through  the  six  fundamental  points.  But  we  have 
supposed  that  K  is  of  higher  order  than  the  third  (see  §  8),  hence 
this  exceptional  case  can  not  take  place,  and  there  exists  therefore 
only  a  finite  number  of  points  P^on  K'  in  which  the  tangent  to  K' 
lies  in  a  plane  of  the  pencil  (39).  For  each  such  point  P2  draw 
the  line  P[P2,  it  meets  the  cubic  surface  F3  in  a  third  point  R. 
As  the  point  P[  describes  the  curve  K\  these  points  R  will  de- 
scribe a  finite  number  of  curves  on  FB .  The  totality  of  these  curves 
makes  up  a  locus  which  we  denote  by  A5 .  Hence  if  we  impose  upon 
the  center  of  projection  O,  condition  VI,  that  O  shall  not  lie  on  this 
locus  A5 ,  then  the  new  double  points  D " ,  •  •  • ,  D  "m  will  have  distinct 
tangents. 

We  will  now  give  a  resume"  of  our  results.  There  was  given  in 
the  plane  II  an  irreducible  algebraic  curve  K  (of  order  higher  than 
the  third)  with  r  ordinary  multiple  points  Av ,  A2 ,  •  •  • ,  Ar .  We 
selected  the  six  fundamental  points  of  the  transformation,  so  that  no 
three  lie  on  a  straight  line  and  not  all  six  on  a  conic  ;  we  chose  one 
of  them  at  the  point  A1 ,  the  other  five  external  to  the  curve  K. 
We  thus  obtained  a  one-to-one  correspondence  between  the  curve  K 
and  its  image  K'  on  the  surface  F% .  This  curve  K' ,  we  projected 
from  a  point  0  of  the  surface  Fz  upon  a  second  plane  II' .  We 
imposed  upon  the  point  O  the  conditions  not  to  lie  on  certain  curves, 
viz.,  K'  and  the  loci  denoted  by  At,  A2,  A,,  A4,  A5. 

Then  we  obtained  a  one-to-one  correspondence  between  the  curve 
K'  and  its  image  K"  in  the  plane  IT',  and  the  curve  K"  has,  corre- 
sponding to  the  multiple  points  A2,  •  •  •,  Ari  r  —  1  ordinary  multiple 
points  A'l ,  •  • .,  A"r  of  the  same  respective  multiplicities  as  A%,  -,Ar; 
whereas,  the  multiple  point  A1  has  been  resolved  into  ax  ordinary 
points.  Besides  the  curve  K"  has  a  certain  number  of  new  double 
points  D",  •  •  •,  D"m  with  linear  cycles  and  distinct  tangents. 

The  points  A2 ,  •  •  •,  A!'r\  D'^  •  •  •,  D"m  are  the  only  singular  points 
of  K". 

We  now  apply  to  the  curve  K"  and  to  the  multiple  point  A2  the 
same  process  which  has  been  applied  to  the  curve  K  and  the  multiple 
point  A1 ,  and  so  on.     We  repeat  the  process  as  many  times  as  there 


52  SINGULARITIES   OF   ALGEBRAIC  CURVES. 

are  multiple  points  A1 ,  •  * »,  Ar  to  be  resolved,  taking  each  time  the 
new  curve  as  the  curve  to  be  transformed,  and  finally  a  curve  will 
be  reached  possessing  a  finite  number  of  ordinary  double  points  with 
distinct  tangents,  but  no  other  multiple  points. 

Combining  this  result  with  Noether's  Theorem  mentioned  in 
the  introduction,  we  have  the  final  theorem  that  every  algebraic 
curve  can  be  transformed,  by  a  birational  transformation  of  the 
curve,  into  a  curve  which  has  no  other  multiple  voints  but  ordinary 
double  points. 


OF  THE 

UNIVERSITY 

OF 


AUTOBIOGRAPHY 

I  was  born  on  the  20th  of  August,  1863,  in  Oktibbeha  County, 
Mississippi.  I  attended  the  public  schools  of  this  county  for  four 
years,  the  High  School  for  four  years,  and  the  Mississippi  Agri- 
cultural and  Mechanical  College  for  three  years,  graduating  in 
1883  with  the  degree  of  B.S.,  and  in  1886  received  the  degree 
of  M.S.  I  attended  the  University  of  Virginia,  the  summers  of 
1885-86-87  ;  the  Universities  of  Gottingen  and  Berlin  the  sem- 
esters of  1888-89.  In  1895  I  entered  the  University  of  Chicago 
as  a  student  of  mathematics  and  astronomy,  and  studied  twelve 
quarters.  I  was  assistant  in  mathematics  in  the  Mississippi  A.  & 
M.  College  from  1883-1888,  Professor  and  Head  of  the  depart- 
ment from  1888-,  and  Director  of  the  School  of  Engineering  from 
1902-.  I  received  instruction  in  mathematics  from  Professors 
Thornton,  Schoenflies,  Schwarz,  Hensel,  Fuchs,  Moore,  Bolza,  and 
Maschke,  and  in  astronomy  from  Professors  Laves  and  Moulton.  My 
thesis  was  carried  on  with  Professor  Bolza.  I  feel  under  a  deep 
obligation  to  all  the  Professors  named,  but  especially  to  Professor 
Bolza  for  the  continued  and  varied  assistance  given  me  throughout 
my  whole  time  of  graduate  work. 

Buz  M.  Walker. 


53 


CONTENTS. 

Introduction 3 

Chapter  I.     Properties  of  Triply  Infinite  Linear  Systems  of 

Plane  Cnbics *        •  6 

Chapter  II.      The  One-to-One  Correspondence  Between  the 

Cubic  Surface  and  the  Plane      .         .         .         .  .16 

Chapter  III.     The  Transformation  of    an  Algebraic  Curve 

from  the  Plane  II  Upon  the  Cubic  Surface  .         .         .27 

Chapter  IV.    Transformation  from  the  Surface  F%  to  a  Plane  IF     37 
Autobiography 53 


54 


AN  tNITIAL  ^5TOCS 

tfuu  BE  ASSESSED  FO«FA.UUBEhe  ^ 

THIS    BOOK  ON   THE  DATEOUE  FOOBTH 

X  .NCBEASETO  SO  «NTSEOgEvENTH     ^ 

DAY    AND    1 
OVERDUE. 


